Integrand size = 24, antiderivative size = 460 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]
3/2*a*arccosh(a*x)^2*(a*x-1)^(1/2)/x/(-a*x+1)^(1/2)-6*a^2*arccosh(a*x)*arc tan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+a^2*arcc osh(a*x)^3*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1)^ (1/2)+3*I*a^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2 )/(-a*x+1)^(1/2)-3/2*I*a^2*arccosh(a*x)^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)* (a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*polylog(2,I*(a*x+(a*x -1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+3/2*I*a^2*arccosh(a *x)^2*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1 )^(1/2)+3*I*a^2*arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2) ))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*arccosh(a*x)*polylog(3,I*(a*x+(a*x -1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*polylog(4,- I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+3*I*a^2* polylog(4,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2 )-1/2*arccosh(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1051\) vs. \(2(460)=920\).
Time = 4.23 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.28 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {i a^2 (1+a x) \left (7 \pi ^4 \sqrt {\frac {-1+a x}{1+a x}}+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+24 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+\frac {192 i \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2}{a x}+\frac {64 i (-1+a x) \text {arccosh}(a x)^3}{a^2 x^2}-32 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3-16 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^4-384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (1+i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-96 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-64 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )+96 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )-8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (1+i e^{\text {arccosh}(a x)}\right )+64 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arccosh}(a x))\right )\right )-48 \sqrt {\frac {-1+a x}{1+a x}} \left (8+\pi ^2-4 i \pi \text {arccosh}(a x)-4 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+192 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (4,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )}{128 \sqrt {1-a^2 x^2}} \]
((-1/128*I)*a^2*(1 + a*x)*(7*Pi^4*Sqrt[(-1 + a*x)/(1 + a*x)] + (8*I)*Pi^3* Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + 24*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x )]*ArcCosh[a*x]^2 + ((192*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2)/(a *x) + ((64*I)*(-1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2) - (32*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x ]^4 - 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x] ] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I/E^ArcCosh[a*x]] + 384* Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^ 2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - (96* I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] + (96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh [a*x]] - (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I*E^ArcCosh[a*x]] + 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4] ] - 48*Sqrt[(-1 + a*x)/(1 + a*x)]*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*Ar cCosh[a*x]^2)*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a *x)]*PolyLog[2, I/E^ArcCosh[a*x]] + 192*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh [a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + ...
Time = 2.50 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.59, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6347, 6298, 6361, 3042, 4668, 3011, 6362, 3042, 4668, 2715, 2838, 7163, 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.
\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6347 |
\(\displaystyle \frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {3 a \sqrt {a x-1} \int \frac {\text {arccosh}(a x)^2}{x^2}dx}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6361 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \int \frac {\text {arccosh}(a x)^3}{a x}d\text {arccosh}(a x)}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \int \text {arccosh}(a x)^3 \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (-3 i \int \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+3 i \int \text {arccosh}(a x)^2 \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6362 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (2 a \int \frac {\text {arccosh}(a x)}{a x}d\text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{x}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \int \text {arccosh}(a x) \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+i \int \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int e^{-\text {arccosh}(a x)} \log \left (1-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+i \int e^{-\text {arccosh}(a x)} \log \left (1+i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {a^2 \sqrt {a x-1} \left (2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {3 a \sqrt {a x-1} \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )}{2 \sqrt {1-a x}}\) |
-1/2*(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/x^2 - (3*a*Sqrt[-1 + a*x]*(-(ArcCo sh[a*x]^2/x) + 2*a*(2*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]] - I*PolyLog[2, ( -I)*E^ArcCosh[a*x]] + I*PolyLog[2, I*E^ArcCosh[a*x]])))/(2*Sqrt[1 - a*x]) + (a^2*Sqrt[-1 + a*x]*(2*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] + (3*I)*(-( ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[ 3, (-I)*E^ArcCosh[a*x]] - PolyLog[4, (-I)*E^ArcCosh[a*x]])) - (3*I)*(-(Arc Cosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, I*E ^ArcCosh[a*x]] - PolyLog[4, I*E^ArcCosh[a*x]]))))/(2*Sqrt[1 - a*x])
3.3.60.3.1 Defintions of rubi rules used
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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[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/(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_.)*(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[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[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[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{3} \sqrt {-a^{2} x^{2}+1}}d x\]
\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]