Integrand size = 29, antiderivative size = 597 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \text {arccosh}(c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
1/3*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2/d^2/(-c^2*d*x^2+d) ^(1/2)+(a+b*arccosh(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*b*c*x*(a+b*arccos h(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)+ 2*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/ 2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+14/3*b*(a+b*arccosh(c*x))*arctan h(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d *x^2+d)^(1/2)+7/3*b^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^ (1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-2*I*b*(a+b*arccosh(c*x))*poly log(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^ 2/(-c^2*d*x^2+d)^(1/2)+2*I*b*(a+b*arccosh(c*x))*polylog(2,I*(c*x+(c*x-1)^( 1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)- 7/3*b^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^( 1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*polylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x +1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*p olylog(3,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/ d^2/(-c^2*d*x^2+d)^(1/2)
Time = 9.53 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {a^2 \log (c x)}{d^{5/2}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{5/2}}+\frac {a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (14 \text {arccosh}(c x) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {csch}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \text {csch}^4\left (\frac {1}{2} \text {arccosh}(c x)\right )-24 i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+24 i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+28 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-28 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-24 i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+24 i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-\text {sech}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {8 \text {arccosh}(c x) \sinh ^4\left (\frac {1}{2} \text {arccosh}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}-14 \text {arccosh}(c x) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{12 d^2 \sqrt {-d (-1+c x) (1+c x)}}+\frac {b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-4 \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )+14 \text {arccosh}(c x)^2 \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-2 \text {arccosh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(c x)\right )-56 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )-24 i \text {arccosh}(c x)^2 \log \left (1-i e^{-\text {arccosh}(c x)}\right )+24 i \text {arccosh}(c x)^2 \log \left (1+i e^{-\text {arccosh}(c x)}\right )+56 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )-56 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-48 i \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+48 i \text {arccosh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+56 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )-48 i \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(c x)}\right )+48 i \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(c x)}\right )-2 \text {arccosh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {8 \text {arccosh}(c x)^2 \sinh ^4\left (\frac {1}{2} \text {arccosh}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+4 \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )-14 \text {arccosh}(c x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{24 d^2 \sqrt {-d (-1+c x) (1+c x)}} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c ^2*x^2))) + (a^2*Log[c*x])/d^(5/2) - (a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c ^2*x^2))]])/d^(5/2) + (a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcCos h[c*x]*Coth[ArcCosh[c*x]/2] - Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Csch[ArcCosh[c*x]/2]^4)/2 - (24*I)*ArcCosh [c*x]*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[ c*x]] + 28*Log[Cosh[ArcCosh[c*x]/2]] - 28*Log[Sinh[ArcCosh[c*x]/2]] - (24* I)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (24*I)*PolyLog[2, I/E^ArcCosh[c*x]] - Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) - 14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]) )/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) + (b^2*Sqrt[(-1 + c*x)/(1 + c*x )]*(1 + c*x)*(-4*Coth[ArcCosh[c*x]/2] + 14*ArcCosh[c*x]^2*Coth[ArcCosh[c*x ]/2] - 2*ArcCosh[c*x]*Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)] *(1 + c*x)*ArcCosh[c*x]^2*Csch[ArcCosh[c*x]/2]^4)/2 - 56*ArcCosh[c*x]*Log[ 1 - E^(-ArcCosh[c*x])] - (24*I)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 56*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 56*PolyLog[2, -E^(-ArcCosh[c*x])] - (48*I)*ArcCosh[c *x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (48*I)*ArcCosh[c*x]*PolyLog[2, I/E^A rcCosh[c*x]] + 56*PolyLog[2, E^(-ArcCosh[c*x])] - (48*I)*PolyLog[3, (-I)/E ^ArcCosh[c*x]] + (48*I)*PolyLog[3, I/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*S...
Time = 4.99 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.75, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.862, Rules used = {6351, 6304, 6316, 83, 6318, 3042, 26, 4670, 2715, 2838, 6351, 25, 6304, 6318, 3042, 26, 4670, 2715, 2838, 6361, 3042, 4668, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6361 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(a+b \text {arccosh}(c x))^2}{c x}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))^2 \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
(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/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCos h[c*x]))/(2*(1 - c^2*x^2)) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh [c*x]]))/c))/(3*d^2*Sqrt[d - c^2*d*x^2]) + ((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]*((2*I)*(a + b*Arc Cosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b *PolyLog[2, E^ArcCosh[c*x]]))/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(2*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]] + (2*I)*b*(-( (a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]]) + b*PolyLog[3, (-I)* E^ArcCosh[c*x]]) - (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[ c*x]]) + b*PolyLog[3, I*E^ArcCosh[c*x]])))/(d*Sqrt[d - c^2*d*x^2]))/d
3.3.20.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[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_.)*((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])
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x ]/Sqrt[d + e*x^2])] Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] , x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int egerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
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^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
-1/3*a^2*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2 ) - 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1/((-c^2*d*x^2 + d)^(3/2)*d)) + integra te(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(5/2)*x) + 2*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x) , x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]