Integrand size = 29, antiderivative size = 476 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
-(a+b*arccosh(c*x))^2/d/x/(-c^2*d*x^2+d)^(3/2)+4/3*c^2*x*(a+b*arccosh(c*x) )^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*c^2*x/d^2/(-c^2*d*x^2+d)^(1/2)+8/3*c^2* 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/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)+8/3*c*( 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)-4 *b*c*(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)-16/3*b*c*(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)-b^2*c*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)-5/3*b^2*c*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 = 2.85 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {c \left (\frac {a^2 \left (3-12 c^2 x^2+8 c^4 x^4\right )}{c x \left (-1+c^2 x^2\right )}+a b \left (10 c x \text {arccosh}(c x)-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+2 c x \text {arccosh}(c x)}{-1+c^2 x^2}-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\frac {3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{c x}+3 \log (c x)+5 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )+b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {\text {arccosh}(c x)}{1-c^2 x^2}-8 \text {arccosh}(c x)^2-\frac {c x \text {arccosh}(c x)^2}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+\frac {5 c x \text {arccosh}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{c x}-10 \text {arccosh}(c x) \log \left (1-e^{-2 \text {arccosh}(c x)}\right )-6 \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+5 \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
(c*((a^2*(3 - 12*c^2*x^2 + 8*c^4*x^4))/(c*x*(-1 + c^2*x^2)) + a*b*(10*c*x* ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCosh[c*x]) /(-1 + c^2*x^2) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((-3*Sqrt[(-1 + c *x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + 3*Log[c*x] + 5*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])) + b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) *((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + ArcCosh[c*x]/(1 - c^2*x^2) - 8*ArcCosh[c*x]^2 - (c*x*ArcCosh[c*x]^2)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) + (5*c*x*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c*x) - 10*ArcC osh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 6*ArcCosh[c*x]*Log[1 + E^(-2*ArcCo sh[c*x])] + 3*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 5*PolyLog[2, E^(-2*ArcCos h[c*x])])))/(3*d^2*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 4.14 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {6347, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6331 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
-((a + b*ArcCosh[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2))) - (2*b*c*Sqrt[-1 + c *x]*Sqrt[1 + c*x]*(-1/2*(b*c*x)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*Ar cCosh[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*PolyLo g[2, E^(2*ArcCosh[c*x])])))/(d^2*Sqrt[d - c^2*d*x^2]) + 4*c^2*((x*(a + b*A rcCosh[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*ArcC osh[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*Sqrt[d - c^2*d*x^2])))/(3*d))
3.3.21.3.1 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[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[(((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. \(2856\) vs. \(2(470)=940\).
Time = 1.50 (sec) , antiderivative size = 2857, normalized size of antiderivative = 6.00
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2857\) |
| parts | \(\text {Expression too large to display}\) | \(2857\) |
-64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x ^5*arccosh(c*x)^2*c^6-280/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c ^4*x^4+26*c^2*x^2-9)*x^5*arccosh(c*x)*c^6+56*b^2*(-d*(c^2*x^2-1))^(1/2)/d^ 3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^3*arccosh(c*x)^2*c^4+48*b^2*(-d*(c ^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^3*arccosh(c*x)* c^4-44*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)* x*arccosh(c*x)^2*c^2-8*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^ 4+26*c^2*x^2-9)*x*arccosh(c*x)*c^2-3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6 *x^6-25*c^4*x^4+26*c^2*x^2-9)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c-64/3*b^2*(-d*( c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^9*arccosh(c*x) *c^10-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(16*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+16*arccosh(c*x)*c^5*x^5-6*ln(1 +(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^5*c^5-10*ln((c*x+(c*x-1)^(1/2)*(c* x+1)^(1/2))^2-1)*x^5*c^5-24*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x ^2-32*c^3*x^3*arccosh(c*x)+12*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^ 3*c^3+20*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^3-c^3*x^3+6*arcco sh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+16*c*x*arccosh(c*x)-6*ln(1+(c*x+(c*x-1 )^(1/2)*(c*x+1)^(1/2))^2)*x*c-10*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1) *x*c+c*x)/d^3/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/x-88/3*b^2*(-d*(c^2*x^2-1))^ (1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^5*(c*x-1)*(c*x+1)*c^6+8...
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
1/3*a^2*(8*c^2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 4*c^2*x/((-c^2*d*x^2 + d)^(3 /2)*d) - 3/((-c^2*d*x^2 + d)^(3/2)*d*x)) + integrate(b^2*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(5/2)*x^2) + 2*a*b*log(c*x + sqr t(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]