\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx\) [237]

Optimal result

Integrand size = 26, antiderivative size = 277 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^5 d} \] Output:

-22/9*b^2*x/c^4/d+2/27*b^2*x^3/c^2/d+22/9*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh 
(c*x))/c^5/d-2/9*b*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3/d-x*(a+b*a 
rcsinh(c*x))^2/c^4/d+1/3*x^3*(a+b*arcsinh(c*x))^2/c^2/d+2*(a+b*arcsinh(c*x 
))^2*arctan(c*x+(c^2*x^2+1)^(1/2))/c^5/d-2*I*b*(a+b*arcsinh(c*x))*polylog( 
2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d+2*I*b*(a+b*arcsinh(c*x))*polylog(2,I*( 
c*x+(c^2*x^2+1)^(1/2)))/c^5/d+2*I*b^2*polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)) 
)/c^5/d-2*I*b^2*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {-3 a^2 c x+a^2 c^3 x^3+3 a^2 \arctan (c x)-\frac {2}{3} a b \left (-11 \sqrt {1+c^2 x^2}+c^2 x^2 \sqrt {1+c^2 x^2}+9 c x \text {arcsinh}(c x)-3 c^3 x^3 \text {arcsinh}(c x)-9 i \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+9 i \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+9 i \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-9 i \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+3 b^2 \left (\frac {5}{2} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-\frac {5}{4} c x \left (2+\text {arcsinh}(c x)^2\right )-\frac {1}{18} \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x))+i \left (-\text {arcsinh}(c x)^2 \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )-2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )+\frac {1}{108} \left (2+9 \text {arcsinh}(c x)^2\right ) \sinh (3 \text {arcsinh}(c x))\right )}{3 c^5 d} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]
 

Output:

(-3*a^2*c*x + a^2*c^3*x^3 + 3*a^2*ArcTan[c*x] - (2*a*b*(-11*Sqrt[1 + c^2*x 
^2] + c^2*x^2*Sqrt[1 + c^2*x^2] + 9*c*x*ArcSinh[c*x] - 3*c^3*x^3*ArcSinh[c 
*x] - (9*I)*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (9*I)*ArcSinh[c*x]*Lo 
g[1 + I*E^ArcSinh[c*x]] + (9*I)*PolyLog[2, (-I)*E^ArcSinh[c*x]] - (9*I)*Po 
lyLog[2, I*E^ArcSinh[c*x]]))/3 + 3*b^2*((5*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]) 
/2 - (5*c*x*(2 + ArcSinh[c*x]^2))/4 - (ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]])/ 
18 + I*(-(ArcSinh[c*x]^2*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[ 
c*x]])) - 2*ArcSinh[c*x]*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2, I/E 
^ArcSinh[c*x]]) - 2*PolyLog[3, (-I)/E^ArcSinh[c*x]] + 2*PolyLog[3, I/E^Arc 
Sinh[c*x]]) + ((2 + 9*ArcSinh[c*x]^2)*Sinh[3*ArcSinh[c*x]])/108))/(3*c^5*d 
)
 

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6227, 27, 6227, 15, 6204, 3042, 4668, 3011, 2720, 6213, 24, 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.

\(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d x^2+d} \, dx\)

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{c^2 d}-\frac {2 b \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 6204

\(\displaystyle -\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {-\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {-\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {-\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 6213

\(\displaystyle -\frac {-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}\)

Input:

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]
 

Output:

(x^3*(a + b*ArcSinh[c*x])^2)/(3*c^2*d) - (2*b*(-1/9*(b*x^3)/c + (x^2*Sqrt[ 
1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) - (2*(-((b*x)/c) + (Sqrt[1 + c^ 
2*x^2]*(a + b*ArcSinh[c*x]))/c^2))/(3*c^2)))/(3*c*d) - ((x*(a + b*ArcSinh[ 
c*x])^2)/c^2 - (2*b*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])) 
/c^2))/c - (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-(( 
a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + b*PolyLog[3, (-I)*E 
^ArcSinh[c*x]]) - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c 
*x]]) + b*PolyLog[3, I*E^ArcSinh[c*x]]))/c^3)/(c^2*d)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 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 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

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]
 

Maple [F]

\[\int \frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{c^{2} d \,x^{2}+d}d x\]

Input:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d),x)
 

Output:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d),x)
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} + d} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d),x, algorithm="fricas")
 

Output:

integral((b^2*x^4*arcsinh(c*x)^2 + 2*a*b*x^4*arcsinh(c*x) + a^2*x^4)/(c^2* 
d*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \] Input:

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d),x)
 

Output:

(Integral(a**2*x**4/(c**2*x**2 + 1), x) + Integral(b**2*x**4*asinh(c*x)**2 
/(c**2*x**2 + 1), x) + Integral(2*a*b*x**4*asinh(c*x)/(c**2*x**2 + 1), x)) 
/d
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} + d} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d),x, algorithm="maxima")
 

Output:

1/3*a^2*((c^2*x^3 - 3*x)/(c^4*d) + 3*arctan(c*x)/(c^5*d)) + integrate(b^2* 
x^4*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^2 + d) + 2*a*b*x^4*log(c*x + s 
qrt(c^2*x^2 + 1))/(c^2*d*x^2 + d), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \] Input:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2),x)
 

Output:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {3 \mathit {atan} \left (c x \right ) a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{2} x^{2}+1}d x \right ) a b \,c^{5}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{2} x^{2}+1}d x \right ) b^{2} c^{5}+a^{2} c^{3} x^{3}-3 a^{2} c x}{3 c^{5} d} \] Input:

int(x^4*(a+b*asinh(c*x))^2/(c^2*d*x^2+d),x)
 

Output:

(3*atan(c*x)*a**2 + 6*int((asinh(c*x)*x**4)/(c**2*x**2 + 1),x)*a*b*c**5 + 
3*int((asinh(c*x)**2*x**4)/(c**2*x**2 + 1),x)*b**2*c**5 + a**2*c**3*x**3 - 
 3*a**2*c*x)/(3*c**5*d)