Integrand size = 29, antiderivative size = 496 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
1/3*b^2*c^2*(-c*x+1)*(c*x+1)/d/x/(-c^2*d*x^2+d)^(1/2)-1/3*(a+b*arccosh(c*x ))^2/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*c^2*(a+b*arccosh(c*x))^2/d/x/(-c^2*d*x ^2+d)^(1/2)+8/3*c^4*x*(a+b*arccosh(c*x))^2/d/(-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/x^2/(-c^2*d*x^2+d)^(1/2)+ 8/3*c^3*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^ (1/2)-20/3*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/(-c^2*d*x^2+d)^(1/2)-16/3*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/(-c^2*d*x^2+d)^(1/2)-5/3*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/(-c^2*d*x^2+d)^(1/2)-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/(-c^2*d*x^2+d)^(1/2)
Time = 1.97 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a^2 \left (-1-4 c^2 x^2+8 c^4 x^4\right )+a b \left (6 c^4 x^4 \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (c x+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)\right )+2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (5 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (5 \log (c x)+3 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )\right )+b^2 \left (c^2 x^2-c^4 x^4+c x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)+3 c^4 x^4 \text {arccosh}(c x)^2+(-1+c x) (1+c x) \text {arccosh}(c x)^2+5 c^2 x^2 (-1+c x) (1+c x) \text {arccosh}(c x)^2-8 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2-6 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-2 \text {arccosh}(c x)}\right )-10 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+5 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+3 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 d x^3 \sqrt {d-c^2 d x^2}} \]
(a^2*(-1 - 4*c^2*x^2 + 8*c^4*x^4) + a*b*(6*c^4*x^4*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(c*x + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) *ArcCosh[c*x]) + 2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(5*Sqrt[(- 1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(5*Log[c*x] + 3*Log[Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)]))) + b^2*(c^2*x^2 - c^4*x^4 + c*x*Sqrt[(- 1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 3*c^4*x^4*ArcCosh[c*x]^2 + (- 1 + c*x)*(1 + c*x)*ArcCosh[c*x]^2 + 5*c^2*x^2*(-1 + c*x)*(1 + c*x)*ArcCosh [c*x]^2 - 8*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 - 6*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-2* ArcCosh[c*x])] - 10*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c *x]*Log[1 + E^(-2*ArcCosh[c*x])] + 5*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 3*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*d*x^3*Sqrt[d - c^2*d* x^2])
Result contains complex when optimal does not.
Time = 4.18 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.95, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {6347, 25, 6327, 6347, 25, 106, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {4}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{x^3 (1-c x) (c x+1)}dx}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3 (1-c x) (c x+1)}dx}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {4}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{x (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 106 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {4}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {4}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} c^2 \left (2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {4}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {4}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {4}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6331 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 c^2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 c^2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {4 i b c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i c^2 \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {4 i b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i c^2 \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {4}{3} c^2 \left (\frac {4 i b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i c^2 \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-2 i c^2 \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{2 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {4}{3} c^2 \left (\frac {4 i b c \sqrt {c x-1} \sqrt {c x+1} \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}\) |
-1/3*(a + b*ArcCosh[c*x])^2/(d*x^3*Sqrt[d - c^2*d*x^2]) - (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (a + b*Arc Cosh[c*x])/(2*x^2) - (2*I)*c^2*(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])])))/(3*d*Sqrt[d - c^2*d*x^2]) + (4*c^2*(-((a + b*ArcCosh[ c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + ((4*I)*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c* x]*(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*PolyLog[2, E^(2*ArcCosh[c*x])]))/(d*Sqrt[ d - c^2*d*x^2]) + 2*c^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*Sqrt[d - c^2*d*x^2]))))/3
3.3.13.3.1 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_.) + (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] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
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_.)*((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)), 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]
Leaf count of result is larger than twice the leaf count of optimal. \(2162\) vs. \(2(486)=972\).
Time = 1.40 (sec) , antiderivative size = 2163, normalized size of antiderivative = 4.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2163\) |
| parts | \(\text {Expression too large to display}\) | \(2163\) |
-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((c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-10*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1 )^(1/2))^2)*x^5*c^5-8*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-16* c^3*x^3*arccosh(c*x)+6*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^3+1 0*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+c^3*x^3-2*arccosh(c*x) *(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*x)/d^2/(c^4*x^4-2*c^2*x^2+1)/x^3-32/3*b^2*( -d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^7*c^10+40/3*b^2*(-d*(c ^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^5*c^8-7/3*b^2*(-d*(c^2*x^2- 1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2) /d^2/(8*c^4*x^4-7*c^2*x^2-1)/x*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c ^4*x^4-7*c^2*x^2-1)/x^3*arccosh(c*x)^2-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/ (8*c^4*x^4-7*c^2*x^2-1)/x^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c+10/ 3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*a rccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3+2*b^2*(-d*(c^2*x ^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1 -c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^3-32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/ (8*c^4*x^4-7*c^2*x^2-1)*x^3*arccosh(c*x)*(c*x-1)*(c*x+1)*c^6+64/3*b^2*(-d* (c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^2*arccosh(c*x)^2*(c*x-1)^ (1/2)*(c*x+1)^(1/2)*c^5-8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7...
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a ^2)/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a^2 + integrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(3/2)*x^4) + 2*a*b*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(3/2)*x^4), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]