\(\int x (a+a \cosh (x))^{3/2} \, dx\) [135]

Optimal result

Integrand size = 12, antiderivative size = 89 \[ \int x (a+a \cosh (x))^{3/2} \, dx=-\frac {16}{3} a \sqrt {a+a \cosh (x)}-\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {8}{3} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63 \[ \int x (a+a \cosh (x))^{3/2} \, dx=\frac {1}{9} a \sqrt {a (1+\cosh (x))} \text {sech}\left (\frac {x}{2}\right ) \left (-54 \cosh \left (\frac {x}{2}\right )-2 \cosh \left (\frac {3 x}{2}\right )+3 x \left (9 \sinh \left (\frac {x}{2}\right )+\sinh \left (\frac {3 x}{2}\right )\right )\right ) \] Input:

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

Output:

(a*Sqrt[a*(1 + Cosh[x])]*Sech[x/2]*(-54*Cosh[x/2] - 2*Cosh[(3*x)/2] + 3*x* 
(9*Sinh[x/2] + Sinh[(3*x)/2])))/9
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 3800, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

Input:

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

Output:

2*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*((-4*Cosh[x/2]^3)/9 + (2*x*Cosh[x/2]^2*S 
inh[x/2])/3 + (2*(-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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x*(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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int x (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{18} \, {\left (3 \, \sqrt {2} a^{\frac {3}{2}} x + 2 \, \sqrt {2} a^{\frac {3}{2}} - {\left (3 \, \sqrt {2} a^{\frac {3}{2}} x - 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 27 \, {\left (\sqrt {2} a^{\frac {3}{2}} x - 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 27 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \] Input:

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

Output:

-1/18*(3*sqrt(2)*a^(3/2)*x + 2*sqrt(2)*a^(3/2) - (3*sqrt(2)*a^(3/2)*x - 2* 
sqrt(2)*a^(3/2))*e^(3*x) - 27*(sqrt(2)*a^(3/2)*x - 2*sqrt(2)*a^(3/2))*e^(2 
*x) + 27*(sqrt(2)*a^(3/2)*x + 2*sqrt(2)*a^(3/2))*e^x)*e^(-3/2*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int x (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{18} \, \sqrt {2} {\left (18 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 3 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 36 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (3 \, a^{\frac {3}{2}} x - 2 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 27 \, {\left (a^{\frac {3}{2}} x - 2 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 9 \, {\left (a^{\frac {3}{2}} x + 2 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \] Input:

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

Output:

-1/18*sqrt(2)*(18*a^(3/2)*x*e^(-1/2*x) + 3*a^(3/2)*x*e^(-3/2*x) + 36*a^(3/ 
2)*e^(-1/2*x) + 2*a^(3/2)*e^(-3/2*x) - (3*a^(3/2)*x - 2*a^(3/2))*e^(3/2*x) 
 - 27*(a^(3/2)*x - 2*a^(3/2))*e^(1/2*x) + 9*(a^(3/2)*x + 2*a^(3/2))*e^(-1/ 
2*x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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