\(\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx\) [47]

Optimal result

Integrand size = 23, antiderivative size = 432 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\frac {2 b^2 (a+b \text {arccosh}(c+d x))^2}{d e^4 (c+d x)}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arccosh}(c+d x))^3 \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {2 i b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {4 i b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {4 i b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(c+d x)}\right )}{d e^4} \] Output:

2*b^2*(a+b*arccosh(d*x+c))^2/d/e^4/(d*x+c)+2/3*b*(d*x+c-1)^(1/2)*(d*x+c+1) 
^(1/2)*(a+b*arccosh(d*x+c))^3/d/e^4/(d*x+c)^2-1/3*(a+b*arccosh(d*x+c))^4/d 
/e^4/(d*x+c)^3-8*b^3*(a+b*arccosh(d*x+c))*arctan(d*x+c+(d*x+c-1)^(1/2)*(d* 
x+c+1)^(1/2))/d/e^4+4/3*b*(a+b*arccosh(d*x+c))^3*arctan(d*x+c+(d*x+c-1)^(1 
/2)*(d*x+c+1)^(1/2))/d/e^4+4*I*b^4*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d* 
x+c+1)^(1/2)))/d/e^4-2*I*b^2*(a+b*arccosh(d*x+c))^2*polylog(2,-I*(d*x+c+(d 
*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-4*I*b^4*polylog(2,I*(d*x+c+(d*x+c-1) 
^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+2*I*b^2*(a+b*arccosh(d*x+c))^2*polylog(2,I* 
(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+4*I*b^3*(a+b*arccosh(d*x+c) 
)*polylog(3,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-4*I*b^3*(a+b 
*arccosh(d*x+c))*polylog(3,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^ 
4-4*I*b^4*polylog(4,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+4*I* 
b^4*polylog(4,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1198\) vs. \(2(432)=864\).

Time = 5.53 (sec) , antiderivative size = 1198, normalized size of antiderivative = 2.77 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^4,x]
 

Output:

(-(a^4/(c + d*x)^3) + 2*a^3*b*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c 
+ d*x))/(c + d*x)^2 - (2*ArcCosh[c + d*x])/(c + d*x)^3 + 2*ArcTan[Tanh[Arc 
Cosh[c + d*x]/2]]) + 6*a^2*b^2*((c + d*x)^(-1) + (Sqrt[(-1 + c + d*x)/(1 + 
 c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x)^2 - ArcCosh[c + d*x]^ 
2/(c + d*x)^3 - I*ArcCosh[c + d*x]*Log[1 - I/E^ArcCosh[c + d*x]] + I*ArcCo 
sh[c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - I*PolyLog[2, (-I)/E^ArcCosh[c 
+ d*x]] + I*PolyLog[2, I/E^ArcCosh[c + d*x]]) + 2*a*b^3*((6*ArcCosh[c + d* 
x])/(c + d*x) + (3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCos 
h[c + d*x]^2)/(c + d*x)^2 - (2*ArcCosh[c + d*x]^3)/(c + d*x)^3 + (3*I)*((4 
*I)*ArcTan[E^ArcCosh[c + d*x]] + ArcCosh[c + d*x]^2*Log[1 - I*E^ArcCosh[c 
+ d*x]] - ArcCosh[c + d*x]^2*Log[1 + I*E^ArcCosh[c + d*x]] - 2*ArcCosh[c + 
 d*x]*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] + 2*ArcCosh[c + d*x]*PolyLog[2, 
I*E^ArcCosh[c + d*x]] + 2*PolyLog[3, (-I)*E^ArcCosh[c + d*x]] - 2*PolyLog[ 
3, I*E^ArcCosh[c + d*x]])) + 3*b^4*(((-7*I)/96)*Pi^4 + (Pi^3*ArcCosh[c + d 
*x])/12 - (I/4)*Pi^2*ArcCosh[c + d*x]^2 + (2*ArcCosh[c + d*x]^2)/(c + d*x) 
 - (Pi*ArcCosh[c + d*x]^3)/3 + (2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + 
c + d*x)*ArcCosh[c + d*x]^3)/(3*(c + d*x)^2) + (I/6)*ArcCosh[c + d*x]^4 - 
ArcCosh[c + d*x]^4/(3*(c + d*x)^3) + (4*I)*ArcCosh[c + d*x]*Log[1 - I/E^Ar 
cCosh[c + d*x]] + (Pi^3*Log[1 + I/E^ArcCosh[c + d*x]])/12 - (4*I)*ArcCosh[ 
c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - (I/2)*Pi^2*ArcCosh[c + d*x]*Lo...
 

Rubi [A] (verified)

Time = 2.66 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6411, 27, 6298, 6348, 6298, 6362, 3042, 4668, 2715, 2838, 3011, 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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{e^4 (c+d x)^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c+d x)^4}d(c+d x)}{d e^4}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {4}{3} b \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6348

\(\displaystyle \frac {\frac {4}{3} b \left (-\frac {3}{2} b \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c+d x)^2}d(c+d x)+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {4}{3} b \left (-\frac {3}{2} b \left (2 b \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6362

\(\displaystyle \frac {\frac {4}{3} b \left (-\frac {3}{2} b \left (2 b \int \frac {a+b \text {arccosh}(c+d x)}{c+d x}d\text {arccosh}(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{c+d x}d\text {arccosh}(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \int (a+b \text {arccosh}(c+d x)) \csc \left (i \text {arccosh}(c+d x)+\frac {\pi }{2}\right )d\text {arccosh}(c+d x)\right )+\frac {1}{2} \int (a+b \text {arccosh}(c+d x))^3 \csc \left (i \text {arccosh}(c+d x)+\frac {\pi }{2}\right )d\text {arccosh}(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (-i b \int \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+i b \int \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )\right )+\frac {1}{2} \left (-3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (-i b \int e^{-\text {arccosh}(c+d x)} \log \left (1-i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}+i b \int e^{-\text {arccosh}(c+d x)} \log \left (1+i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )\right )+\frac {1}{2} \left (-3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (-3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (3 i b \left (2 b \int (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \int (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {3}{2} b \left (-\frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}+2 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )\right )+\frac {1}{2} \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\)

Input:

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^4,x]
 

Output:

(-1/3*(a + b*ArcCosh[c + d*x])^4/(c + d*x)^3 + (4*b*((Sqrt[-1 + c + d*x]*S 
qrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(2*(c + d*x)^2) - (3*b*(-((a 
+ b*ArcCosh[c + d*x])^2/(c + d*x)) + 2*b*(2*(a + b*ArcCosh[c + d*x])*ArcTa 
n[E^ArcCosh[c + d*x]] - I*b*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] + I*b*Poly 
Log[2, I*E^ArcCosh[c + d*x]])))/2 + (2*(a + b*ArcCosh[c + d*x])^3*ArcTan[E 
^ArcCosh[c + d*x]] + (3*I)*b*(-((a + b*ArcCosh[c + d*x])^2*PolyLog[2, (-I) 
*E^ArcCosh[c + d*x]]) + 2*b*((a + b*ArcCosh[c + d*x])*PolyLog[3, (-I)*E^Ar 
cCosh[c + d*x]] - b*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])) - (3*I)*b*(-((a 
+ b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]]) + 2*b*((a + b*Ar 
cCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x]] - b*PolyLog[4, I*E^ArcCos 
h[c + d*x]])))/2))/3)/(d*e^4)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2715

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]
 

rule 2720

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]]]
 

rule 2838

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]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

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]
 

rule 6298

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]
 

rule 6348

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) 
*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( 
m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)* 
(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f 
*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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*ArcCos 
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq 
Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 6362

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]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7143

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]
 

rule 7163

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]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{4}}d x\]

Input:

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^4,x)
 

Output:

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^4,x)
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="fricas")
 

Output:

integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2* 
arccosh(d*x + c)^2 + 4*a^3*b*arccosh(d*x + c) + a^4)/(d^4*e^4*x^4 + 4*c*d^ 
3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \] Input:

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e)**4,x)
                                                                                    
                                                                                    
 

Output:

(Integral(a**4/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 
4*x**4), x) + Integral(b**4*acosh(c + d*x)**4/(c**4 + 4*c**3*d*x + 6*c**2* 
d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(4*a*b**3*acosh(c + d 
*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), 
 x) + Integral(6*a**2*b**2*acosh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d 
**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(4*a**3*b*acosh(c + d* 
x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x)) 
/e**4
 

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="maxima")
 

Output:

-1/3*b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^4*e^4*x^3 
 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*a^4/(d^4*e^4*x^3 + 
 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(2/3*(2*(3*(c^3 
 - c)*a*b^3 + (c^3 - c)*b^4 + (3*a*b^3*d^3 + b^4*d^3)*x^3 + 3*(3*a*b^3*c*d 
^2 + b^4*c*d^2)*x^2 + (b^4*c^2 + 3*(c^2 - 1)*a*b^3 + (3*a*b^3*d^2 + b^4*d^ 
2)*x^2 + 2*(3*a*b^3*c*d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) 
+ (3*(3*c^2*d - d)*a*b^3 + (3*c^2*d - d)*b^4)*x)*log(d*x + sqrt(d*x + c + 
1)*sqrt(d*x + c - 1) + c)^3 + 9*(a^2*b^2*d^3*x^3 + 3*a^2*b^2*c*d^2*x^2 + ( 
3*c^2*d - d)*a^2*b^2*x + (c^3 - c)*a^2*b^2 + (a^2*b^2*d^2*x^2 + 2*a^2*b^2* 
c*d*x + (c^2 - 1)*a^2*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + 
sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 + 6*(a^3*b*d^3*x^3 + 3*a^3*b*c* 
d^2*x^2 + (3*c^2*d - d)*a^3*b*x + (c^3 - c)*a^3*b + (a^3*b*d^2*x^2 + 2*a^3 
*b*c*d*x + (c^2 - 1)*a^3*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + 
 sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 
+ c^7*e^4 - c^5*e^4 + (21*c^2*d^5*e^4 - d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 - 
c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 - 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 - 
 10*c^3*d^2*e^4)*x^2 + (d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 
+ (15*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*( 
5*c^4*d^2*e^4 - 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt 
(d*x + c + 1)*sqrt(d*x + c - 1) + (7*c^6*d*e^4 - 5*c^4*d*e^4)*x), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x + c) + a)^4/(d*e*x + c*e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \] Input:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^4,x)
 

Output:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^4, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^4} \, dx =\text {Too large to display} \] Input:

int((a+b*acosh(d*x+c))^4/(d*e*x+c*e)^4,x)
 

Output:

(12*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x* 
*3 + d**4*x**4),x)*a**3*b*c**3*d + 36*int(acosh(c + d*x)/(c**4 + 4*c**3*d* 
x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a**3*b*c**2*d**2*x + 
36*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x** 
3 + d**4*x**4),x)*a**3*b*c*d**3*x**2 + 12*int(acosh(c + d*x)/(c**4 + 4*c** 
3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a**3*b*d**4*x**3 
+ 3*int(acosh(c + d*x)**4/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3 
*x**3 + d**4*x**4),x)*b**4*c**3*d + 9*int(acosh(c + d*x)**4/(c**4 + 4*c**3 
*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b**4*c**2*d**2*x + 
 9*int(acosh(c + d*x)**4/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3* 
x**3 + d**4*x**4),x)*b**4*c*d**3*x**2 + 3*int(acosh(c + d*x)**4/(c**4 + 4* 
c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b**4*d**4*x**3 
 + 12*int(acosh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d* 
*3*x**3 + d**4*x**4),x)*a*b**3*c**3*d + 36*int(acosh(c + d*x)**3/(c**4 + 4 
*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a*b**3*c**2*d 
**2*x + 36*int(acosh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4 
*c*d**3*x**3 + d**4*x**4),x)*a*b**3*c*d**3*x**2 + 12*int(acosh(c + d*x)**3 
/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a*b 
**3*d**4*x**3 + 18*int(acosh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2* 
x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a**2*b**2*c**3*d + 54*int(acosh(c ...