\(\int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx\) [145]

Optimal result

Integrand size = 14, antiderivative size = 248 \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \] Output:

2*x/a/(a+a*cosh(x))^(1/2)+x^2*arctan(exp(1/2*x))*cosh(1/2*x)/a/(a+a*cosh(x 
))^(1/2)-4*arctan(sinh(1/2*x))*cosh(1/2*x)/a/(a+a*cosh(x))^(1/2)-2*I*x*cos 
h(1/2*x)*polylog(2,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+2*I*x*cosh(1/2*x)* 
polylog(2,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+4*I*cosh(1/2*x)*polylog(3,-I 
*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)-4*I*cosh(1/2*x)*polylog(3,I*exp(1/2*x)) 
/a/(a+a*cosh(x))^(1/2)+1/2*x^2*tanh(1/2*x)/a/(a+a*cosh(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (4 x \cosh \left (\frac {x}{2}\right )+i \cosh ^2\left (\frac {x}{2}\right ) \left (16 i \arctan \left (e^{x/2}\right )+x^2 \log \left (1-i e^{x/2}\right )-x^2 \log \left (1+i e^{x/2}\right )-4 x \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+4 x \operatorname {PolyLog}\left (2,i e^{x/2}\right )+8 \operatorname {PolyLog}\left (3,-i e^{x/2}\right )-8 \operatorname {PolyLog}\left (3,i e^{x/2}\right )\right )+x^2 \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \] Input:

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

Output:

(Cosh[x/2]*(4*x*Cosh[x/2] + I*Cosh[x/2]^2*((16*I)*ArcTan[E^(x/2)] + x^2*Lo 
g[1 - I*E^(x/2)] - x^2*Log[1 + I*E^(x/2)] - 4*x*PolyLog[2, (-I)*E^(x/2)] + 
 4*x*PolyLog[2, I*E^(x/2)] + 8*PolyLog[3, (-I)*E^(x/2)] - 8*PolyLog[3, I*E 
^(x/2)]) + x^2*Sinh[x/2]))/(a*(1 + Cosh[x]))^(3/2)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.60, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3800, 3042, 4674, 3042, 4257, 4668, 3011, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \int x^2 \text {sech}\left (\frac {x}{2}\right )dx-4 \int \text {sech}\left (\frac {x}{2}\right )dx+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \int x^2 \csc \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx-4 \int \csc \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \int x^2 \csc \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx-8 \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (-4 i \int x \log \left (1-i e^{x/2}\right )dx+4 i \int x \log \left (1+i e^{x/2}\right )dx+4 x^2 \arctan \left (e^{x/2}\right )\right )-8 \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (4 i \left (2 \int \operatorname {PolyLog}\left (2,-i e^{x/2}\right )dx-2 x \operatorname {PolyLog}\left (2,-i e^{x/2}\right )\right )-4 i \left (2 \int \operatorname {PolyLog}\left (2,i e^{x/2}\right )dx-2 x \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+4 x^2 \arctan \left (e^{x/2}\right )\right )-8 \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (4 i \left (4 \int e^{-x/2} \operatorname {PolyLog}\left (2,-i e^{x/2}\right )de^{x/2}-2 x \operatorname {PolyLog}\left (2,-i e^{x/2}\right )\right )-4 i \left (4 \int e^{-x/2} \operatorname {PolyLog}\left (2,i e^{x/2}\right )de^{x/2}-2 x \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+4 x^2 \arctan \left (e^{x/2}\right )\right )-8 \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (4 x^2 \arctan \left (e^{x/2}\right )+4 i \left (4 \operatorname {PolyLog}\left (3,-i e^{x/2}\right )-2 x \operatorname {PolyLog}\left (2,-i e^{x/2}\right )\right )-4 i \left (4 \operatorname {PolyLog}\left (3,i e^{x/2}\right )-2 x \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )\right )-8 \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+x^2 \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+4 x \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

Input:

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

Output:

(Cosh[x/2]*(-8*ArcTan[Sinh[x/2]] + (4*x^2*ArcTan[E^(x/2)] + (4*I)*(-2*x*Po 
lyLog[2, (-I)*E^(x/2)] + 4*PolyLog[3, (-I)*E^(x/2)]) - (4*I)*(-2*x*PolyLog 
[2, I*E^(x/2)] + 4*PolyLog[3, I*E^(x/2)]))/2 + 4*x*Sech[x/2] + x^2*Sech[x/ 
2]*Tanh[x/2]))/(2*a*Sqrt[a + a*Cosh[x]])
 

Defintions of rubi rules used

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 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])
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, 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 4674

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(a*cosh(x) + a)*x^2/(a^2*cosh(x)^2 + 2*a^2*cosh(x) + a^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

4/27*sqrt(2)*((3*e^(5/2*x) + 8*e^(3/2*x) - 3*e^(1/2*x))/(a^(3/2)*e^(3*x) + 
 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 3*arctan(e^(1/2*x))/a^(3/2 
)) + 36*sqrt(2)*integrate(1/9*x^2*e^(3/2*x)/(a^(3/2)*e^(4*x) + 4*a^(3/2)*e 
^(3*x) + 6*a^(3/2)*e^(2*x) + 4*a^(3/2)*e^x + a^(3/2)), x) + 48*sqrt(2)*int 
egrate(1/9*x*e^(3/2*x)/(a^(3/2)*e^(4*x) + 4*a^(3/2)*e^(3*x) + 6*a^(3/2)*e^ 
(2*x) + 4*a^(3/2)*e^x + a^(3/2)), x) - 4/27*(9*sqrt(2)*x^2 + 12*sqrt(2)*x 
+ 8*sqrt(2))*e^(3/2*x)/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^ 
x + a^(3/2))
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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