\(\int x^2 (a+a \cosh (x))^{3/2} \, dx\) [134]

Optimal result

Integrand size = 14, antiderivative size = 145 \[ \int x^2 (a+a \cosh (x))^{3/2} \, dx=-\frac {32}{3} a x \sqrt {a+a \cosh (x)}-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^2 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {224}{9} a \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cosh (x)} \sinh ^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \] Output:

-32/3*a*x*(a+a*cosh(x))^(1/2)-16/9*a*x*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)+4 
/3*a*x^2*cosh(1/2*x)*(a+a*cosh(x))^(1/2)*sinh(1/2*x)+224/9*a*(a+a*cosh(x)) 
^(1/2)*tanh(1/2*x)+8/3*a*x^2*(a+a*cosh(x))^(1/2)*tanh(1/2*x)+32/27*a*(a+a* 
cosh(x))^(1/2)*sinh(1/2*x)^2*tanh(1/2*x)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.37 \[ \int x^2 (a+a \cosh (x))^{3/2} \, dx=\frac {2}{27} a \sqrt {a (1+\cosh (x))} \left (-156 x+\left (328+45 x^2\right ) \tanh \left (\frac {x}{2}\right )+\cosh (x) \left (-12 x+\left (8+9 x^2\right ) \tanh \left (\frac {x}{2}\right )\right )\right ) \] Input:

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

Output:

(2*a*Sqrt[a*(1 + Cosh[x])]*(-156*x + (328 + 45*x^2)*Tanh[x/2] + Cosh[x]*(- 
12*x + (8 + 9*x^2)*Tanh[x/2])))/27
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3800, 3042, 3792, 3042, 3113, 2009, 3777, 26, 3042, 26, 3777, 3042, 3117}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx+\frac {16}{9} i \int \left (\sinh ^2\left (\frac {x}{2}\right )+1\right )d\left (-i \sinh \left (\frac {x}{2}\right )\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )-4 i \int -i x \sinh \left (\frac {x}{2}\right )dx\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )-4 \int x \sinh \left (\frac {x}{2}\right )dx\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )-4 \int -i x \sin \left (\frac {i x}{2}\right )dx\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )+4 i \int x \sin \left (\frac {i x}{2}\right )dx\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )+4 i \left (2 i x \cosh \left (\frac {x}{2}\right )-2 i \int \cosh \left (\frac {x}{2}\right )dx\right )\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )+4 i \left (2 i x \cosh \left (\frac {x}{2}\right )-2 i \int \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx\right )\right )+\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3117

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {2}{3} x^2 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^2 \sinh \left (\frac {x}{2}\right )+4 i \left (2 i x \cosh \left (\frac {x}{2}\right )-4 i \sinh \left (\frac {x}{2}\right )\right )\right )+\frac {16}{9} i \left (-\frac {1}{3} i \sinh ^3\left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )-\frac {8}{9} x \cosh ^3\left (\frac {x}{2}\right )\right )\)

Input:

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

Output:

2*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*((-8*x*Cosh[x/2]^3)/9 + (2*x^2*Cosh[x/2] 
^2*Sinh[x/2])/3 + (2*((4*I)*((2*I)*x*Cosh[x/2] - (4*I)*Sinh[x/2]) + 2*x^2* 
Sinh[x/2]))/3 + ((16*I)/9)*((-I)*Sinh[x/2] - (I/3)*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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x^2*(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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int x^2 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 12 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}} - {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{2} - 12 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{2} - 4 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{2} + 4 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \] Input:

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

Output:

-1/54*(9*sqrt(2)*a^(3/2)*x^2 + 12*sqrt(2)*a^(3/2)*x + 8*sqrt(2)*a^(3/2) - 
(9*sqrt(2)*a^(3/2)*x^2 - 12*sqrt(2)*a^(3/2)*x + 8*sqrt(2)*a^(3/2))*e^(3*x) 
 - 81*(sqrt(2)*a^(3/2)*x^2 - 4*sqrt(2)*a^(3/2)*x + 8*sqrt(2)*a^(3/2))*e^(2 
*x) + 81*(sqrt(2)*a^(3/2)*x^2 + 4*sqrt(2)*a^(3/2)*x + 8*sqrt(2)*a^(3/2))*e 
^x)*e^(-3/2*x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99 \[ \int x^2 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, \sqrt {2} {\left (54 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {1}{2} \, x\right )} + 9 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {3}{2} \, x\right )} + 216 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 12 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 432 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 8 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (9 \, a^{\frac {3}{2}} x^{2} - 12 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 81 \, {\left (a^{\frac {3}{2}} x^{2} - 4 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 27 \, {\left (a^{\frac {3}{2}} x^{2} + 4 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \] Input:

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

Output:

-1/54*sqrt(2)*(54*a^(3/2)*x^2*e^(-1/2*x) + 9*a^(3/2)*x^2*e^(-3/2*x) + 216* 
a^(3/2)*x*e^(-1/2*x) + 12*a^(3/2)*x*e^(-3/2*x) + 432*a^(3/2)*e^(-1/2*x) + 
8*a^(3/2)*e^(-3/2*x) - (9*a^(3/2)*x^2 - 12*a^(3/2)*x + 8*a^(3/2))*e^(3/2*x 
) - 81*(a^(3/2)*x^2 - 4*a^(3/2)*x + 8*a^(3/2))*e^(1/2*x) + 27*(a^(3/2)*x^2 
 + 4*a^(3/2)*x + 8*a^(3/2))*e^(-1/2*x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(cosh(x) + 1)*cosh(x)*x**2,x) + int(sqrt(cosh(x) + 1)*x 
**2,x))