\(\int x^3 (a+a \cosh (x))^{3/2} \, dx\) [133]

Optimal result

Integrand size = 14, antiderivative size = 185 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1280}{9} a \sqrt {a+a \cosh (x)}-16 a x^2 \sqrt {a+a \cosh (x)}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {640}{9} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \] Output:

-1280/9*a*(a+a*cosh(x))^(1/2)-16*a*x^2*(a+a*cosh(x))^(1/2)-64/27*a*cosh(1/ 
2*x)^2*(a+a*cosh(x))^(1/2)-8/3*a*x^2*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)+32/ 
9*a*x*cosh(1/2*x)*(a+a*cosh(x))^(1/2)*sinh(1/2*x)+4/3*a*x^3*cosh(1/2*x)*(a 
+a*cosh(x))^(1/2)*sinh(1/2*x)+640/9*a*x*(a+a*cosh(x))^(1/2)*tanh(1/2*x)+8/ 
3*a*x^3*(a+a*cosh(x))^(1/2)*tanh(1/2*x)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.38 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\frac {2}{27} a \sqrt {a (1+\cosh (x))} \left (-2 \left (968+117 x^2\right )+3 x \left (328+15 x^2\right ) \tanh \left (\frac {x}{2}\right )+\cosh (x) \left (-2 \left (8+9 x^2\right )+3 x \left (8+3 x^2\right ) \tanh \left (\frac {x}{2}\right )\right )\right ) \] Input:

Integrate[x^3*(a + a*Cosh[x])^(3/2),x]
 

Output:

(2*a*Sqrt[a*(1 + Cosh[x])]*(-2*(968 + 117*x^2) + 3*x*(328 + 15*x^2)*Tanh[x 
/2] + Cosh[x]*(-2*(8 + 9*x^2) + 3*x*(8 + 3*x^2)*Tanh[x/2])))/27
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.92, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.643, Rules used = {3042, 3800, 3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3791, 3042, 3777, 26, 3042, 26, 3118}

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.

\(\displaystyle \int x^3 (a \cosh (x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 \left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x^3 \cosh ^3\left (\frac {x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x^3 \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \int x^3 \cosh \left (\frac {x}{2}\right )dx+\frac {8}{3} \int x \cosh ^3\left (\frac {x}{2}\right )dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \int x^3 \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 i \int -i x^2 \sinh \left (\frac {x}{2}\right )dx\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 \int x^2 \sinh \left (\frac {x}{2}\right )dx\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 \int -i x^2 \sin \left (\frac {i x}{2}\right )dx\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \int x^2 \sin \left (\frac {i x}{2}\right )dx\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \int x \cosh \left (\frac {x}{2}\right )dx\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 i \int -i \sinh \left (\frac {x}{2}\right )dx\right )\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int \sinh \left (\frac {x}{2}\right )dx\right )\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int -i \sin \left (\frac {i x}{2}\right )dx\right )\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )+2 i \int \sin \left (\frac {i x}{2}\right )dx\right )\right )\right )+\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \int x \cosh \left (\frac {x}{2}\right )dx-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \left (2 x \sinh \left (\frac {x}{2}\right )-2 i \int -i \sinh \left (\frac {x}{2}\right )dx\right )-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int \sinh \left (\frac {x}{2}\right )dx\right )-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int -i \sin \left (\frac {i x}{2}\right )dx\right )-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {8}{3} \left (\frac {2}{3} \left (2 x \sinh \left (\frac {x}{2}\right )+2 i \int \sin \left (\frac {i x}{2}\right )dx\right )-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} x^3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {4}{3} x^2 \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )+\frac {8}{3} \left (-\frac {4}{9} \cosh ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\)

Input:

Int[x^3*(a + a*Cosh[x])^(3/2),x]
 

Output:

2*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*((-4*x^2*Cosh[x/2]^3)/3 + (2*x^3*Cosh[x/ 
2]^2*Sinh[x/2])/3 + (8*((-4*Cosh[x/2]^3)/9 + (2*x*Cosh[x/2]^2*Sinh[x/2])/3 
 + (2*(-4*Cosh[x/2] + 2*x*Sinh[x/2]))/3))/3 + (2*(2*x^3*Sinh[x/2] + (6*I)* 
((2*I)*x^2*Cosh[x/2] - (4*I)*(-4*Cosh[x/2] + 2*x*Sinh[x/2]))))/3)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int x^{3} \left (a +\cosh \left (x \right ) a \right )^{\frac {3}{2}}d x\]

Input:

int(x^3*(a+cosh(x)*a)^(3/2),x)
 

Output:

int(x^3*(a+cosh(x)*a)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(a+a*cosh(x))^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\int x^{3} \left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \] Input:

integrate(x**3*(a+a*cosh(x))**(3/2),x)
 

Output:

Integral(x**3*(a*(cosh(x) + 1))**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} + 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 16 \, \sqrt {2} a^{\frac {3}{2}} - {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} - 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 16 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} - 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \] Input:

integrate(x^3*(a+a*cosh(x))^(3/2),x, algorithm="maxima")
 

Output:

-1/54*(9*sqrt(2)*a^(3/2)*x^3 + 18*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2) 
*x + 16*sqrt(2)*a^(3/2) - (9*sqrt(2)*a^(3/2)*x^3 - 18*sqrt(2)*a^(3/2)*x^2 
+ 24*sqrt(2)*a^(3/2)*x - 16*sqrt(2)*a^(3/2))*e^(3*x) - 81*(sqrt(2)*a^(3/2) 
*x^3 - 6*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2)*x - 48*sqrt(2)*a^(3/2))* 
e^(2*x) + 81*(sqrt(2)*a^(3/2)*x^3 + 6*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^( 
3/2)*x + 48*sqrt(2)*a^(3/2))*e^x)*e^(-3/2*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.04 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, \sqrt {2} {\left (54 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {1}{2} \, x\right )} + 9 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {3}{2} \, x\right )} + 324 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {1}{2} \, x\right )} + 18 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {3}{2} \, x\right )} + 1296 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 24 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 2592 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 16 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (9 \, a^{\frac {3}{2}} x^{3} - 18 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 16 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 81 \, {\left (a^{\frac {3}{2}} x^{3} - 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 48 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 27 \, {\left (a^{\frac {3}{2}} x^{3} + 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x + 48 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \] Input:

integrate(x^3*(a+a*cosh(x))^(3/2),x, algorithm="giac")
 

Output:

-1/54*sqrt(2)*(54*a^(3/2)*x^3*e^(-1/2*x) + 9*a^(3/2)*x^3*e^(-3/2*x) + 324* 
a^(3/2)*x^2*e^(-1/2*x) + 18*a^(3/2)*x^2*e^(-3/2*x) + 1296*a^(3/2)*x*e^(-1/ 
2*x) + 24*a^(3/2)*x*e^(-3/2*x) + 2592*a^(3/2)*e^(-1/2*x) + 16*a^(3/2)*e^(- 
3/2*x) - (9*a^(3/2)*x^3 - 18*a^(3/2)*x^2 + 24*a^(3/2)*x - 16*a^(3/2))*e^(3 
/2*x) - 81*(a^(3/2)*x^3 - 6*a^(3/2)*x^2 + 24*a^(3/2)*x - 48*a^(3/2))*e^(1/ 
2*x) + 27*(a^(3/2)*x^3 + 6*a^(3/2)*x^2 + 24*a^(3/2)*x + 48*a^(3/2))*e^(-1/ 
2*x))
 

Mupad [F(-1)]

Timed out. \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\int x^3\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \] Input:

int(x^3*(a + a*cosh(x))^(3/2),x)
 

Output:

int(x^3*(a + a*cosh(x))^(3/2), x)
 

Reduce [F]

\[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right ) x^{3}d x +\int \sqrt {\cosh \left (x \right )+1}\, x^{3}d x \right ) \] Input:

int(x^3*(a+a*cosh(x))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(cosh(x) + 1)*cosh(x)*x**3,x) + int(sqrt(cosh(x) + 1)*x 
**3,x))