Integrand size = 29, antiderivative size = 562 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
-1/3*(a+b*arccosh(c*x))^2/d/x^3/(-c^2*d*x^2+d)^(3/2)-2*c^2*(a+b*arccosh(c* x))^2/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c^4*x*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^ 2+d)^(3/2)+1/3*b^2*c^2/d^2/x/(-c^2*d*x^2+d)^(1/2)-2/3*b^2*c^4*x/d^2/(-c^2* d*x^2+d)^(1/2)+16/3*c^4*x*(a+b*arccosh(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/ 3*b*c*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x^2/(-c^2*x^2+1)/ (-c^2*d*x^2+d)^(1/2)+16/3*c^3*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^( 1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-32/3*b*c^3*(a+b*arccosh(c*x))*arctanh((c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2 +d)^(1/2)-32/3*b*c^3*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1 /2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b^2*c^3*p olylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2) /d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b^2*c^3*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 2.80 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )}{x^3 \left (-1+c^2 x^2\right )}+a b c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {1}{c^2 x^2}+\frac {1}{1-c^2 x^2}+\frac {2 \left (\frac {-1+c x}{1+c x}\right )^{3/2} \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \text {arccosh}(c x)}{c^3 x^3 (-1+c x)^3}-16 \log (c x)-16 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )+b^2 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\frac {\text {arccosh}(c x)}{c^2 x^2}+\frac {\text {arccosh}(c x)}{1-c^2 x^2}-16 \text {arccosh}(c x)^2-\frac {c x \text {arccosh}(c x)^2}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+\frac {8 c x \text {arccosh}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{c^3 x^3}+\frac {8 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{c x}-16 \text {arccosh}(c x) \log \left (1-e^{-2 \text {arccosh}(c x)}\right )-16 \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/(x^3*(-1 + c^2*x^2)) + a* b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(1/(c^2*x^2) + (1 - c^2*x^2)^(- 1) + (2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6* x^6)*ArcCosh[c*x])/(c^3*x^3*(-1 + c*x)^3) - 16*Log[c*x] - 16*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]) + b^2*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c* x)*((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (Sqrt[(-1 + c*x)/(1 + c*x )]*(1 + c*x))/(c*x) + ArcCosh[c*x]/(c^2*x^2) + ArcCosh[c*x]/(1 - c^2*x^2) - 16*ArcCosh[c*x]^2 - (c*x*ArcCosh[c*x]^2)/(((-1 + c*x)/(1 + c*x))^(3/2)*( 1 + c*x)^3) + (8*c*x*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) ) + (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c^3*x^3) + (8*S qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c*x) - 16*ArcCosh[c*x ]*Log[1 - E^(-2*ArcCosh[c*x])] - 16*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x ])] + 8*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 8*PolyLog[2, E^(-2*ArcCosh[c*x] )]))/(3*d^2*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 6.30 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.33, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.966, Rules used = {6347, 6327, 6347, 114, 27, 41, 6316, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 6329, 41, 6351, 41, 6331, 5984, 3042, 26, 4670, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3 (1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {1}{2} b c \int \frac {1}{x^2 (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {1}{2} b c \left (\int \frac {2 c^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {1}{2} b c \left (2 c^2 \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle 2 c^2 \left (4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6331 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-2 i \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle 2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 \left (4 \left (\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}-2 i \left (-\frac {1}{2} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}\right ) c^2-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-2 i \left (i (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )+\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}\right )\right ) c}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right ) c^2-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (2 \left (-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-2 i \left (i (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )+\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}\right )\right ) c^2-\frac {1}{2} b \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right ) c-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}\right ) c}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (-2 i \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2 \left (1-c^2 x^2\right )}-\frac {1}{2} b c \left (\frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 x}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+2 c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}\) |
-1/3*(a + b*ArcCosh[c*x])^2/(d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*(b*c*(1/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2* c^2*x)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))) - (a + b*ArcCosh[c*x])/(2*x^2*(1 - c^2*x^2)) + 2*c^2*(-1/2*(b*c*x)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*A rcCosh[c*x])/(2*(1 - c^2*x^2)) - (2*I)*(I*(a + b*ArcCosh[c*x])*ArcTanh[E^( 2*ArcCosh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcCosh[c*x])] - (I/4)*b*PolyL og[2, E^(2*ArcCosh[c*x])]))))/(3*d^2*Sqrt[d - c^2*d*x^2]) + 2*c^2*(-((a + b*ArcCosh[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2))) - (2*b*c*Sqrt[-1 + c*x]*Sqr t[1 + c*x]*(-1/2*(b*c*x)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c *x])/(2*(1 - c^2*x^2)) - (2*I)*(I*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCos h[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcCosh[c*x])] - (I/4)*b*PolyLog[2, E^ (2*ArcCosh[c*x])])))/(d^2*Sqrt[d - c^2*d*x^2]) + 4*c^2*((x*(a + b*ArcCosh[ c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x] *(-1/2*(b*x)/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(2*c^ 2*(1 - c^2*x^2))))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*((x*(a + b*ArcCosh[c*x ])^2)/(d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(((- 1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2*((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/4)))/(c*d*Sqr t[d - c^2*d*x^2])))/(3*d)))
3.3.23.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp [b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x] ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[-d^(-1) Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x , ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IG tQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp [b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[( f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^ (n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[ (f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x]) ^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] & & GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(3744\) vs. \(2(544)=1088\).
Time = 1.39 (sec) , antiderivative size = 3745, normalized size of antiderivative = 6.66
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3745\) |
| parts | \(\text {Expression too large to display}\) | \(3745\) |
a^2*(-1/3/d/x^3/(-c^2*d*x^2+d)^(3/2)+2*c^2*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4* c^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2))))+128/3* b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^ 2-1)*x^5*(c*x-1)*(c*x+1)*c^8-8*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8- 36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^9+25 6/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^ 2*x^2-1)*x^9*(c*x-1)*(c*x+1)*arccosh(c*x)*c^12-640/3*b^2*(-d*(c^2*x^2-1))^ (1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*(c*x-1)*(c*x +1)*arccosh(c*x)*c^10+64*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6 *x^6+35*c^4*x^4-10*c^2*x^2-1)*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x) ^2*c^9+160*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^ 4-10*c^2*x^2-1)*x^5*(c*x-1)*(c*x+1)*arccosh(c*x)*c^8-128*b^2*(-d*(c^2*x^2- 1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^4*(c*x-1)^ (1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*c^7-80/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3 /(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*(c*x-1)*(c*x+1)*arcco sh(c*x)*c^6+176/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35 *c^4*x^4-10*c^2*x^2-1)*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*c^5+ 8/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1) *polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3+16/3*b^2*(-d*(c^2*x^2 -1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*polylog(2,c*x+(c...
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
1/3*a*b*c*(8*c^2*sqrt(-d)*log(c*x + 1)/d^3 + 8*c^2*sqrt(-d)*log(c*x - 1)/d ^3 + 16*c^2*sqrt(-d)*log(x)/d^3 + sqrt(-d)/(c^2*d^3*x^4 - d^3*x^2)) + 2/3* (16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*a *b*arccosh(c*x) + 1/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^ 2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^ 2 + d)^(3/2)*d*x^3))*a^2 + b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(5/2)*x^4), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]