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

Optimal. Leaf size=402 \[ -\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{3 x^2}{a \sqrt{a \cosh (x)+a}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}} \]

[Out]

(3*x^2)/(a*Sqrt[a + a*Cosh[x]]) - (24*x*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*ArcTan[E^(x/
2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) -
((3*I)*x^2*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[2, I*E^(x/
2)])/(a*Sqrt[a + a*Cosh[x]]) + ((3*I)*x^2*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((12*I)*x
*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((12*I)*x*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*S
qrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[4, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*
PolyLog[4, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*Tanh[x/2])/(2*a*Sqrt[a + a*Cosh[x]])

________________________________________________________________________________________

Rubi [A]  time = 0.257355, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {3319, 4186, 4180, 2279, 2391, 2531, 6609, 2282, 6589} \[ -\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{3 x^2}{a \sqrt{a \cosh (x)+a}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + a*Cosh[x])^(3/2),x]

[Out]

(3*x^2)/(a*Sqrt[a + a*Cosh[x]]) - (24*x*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*ArcTan[E^(x/
2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) -
((3*I)*x^2*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[2, I*E^(x/
2)])/(a*Sqrt[a + a*Cosh[x]]) + ((3*I)*x^2*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((12*I)*x
*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((12*I)*x*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*S
qrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[4, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*
PolyLog[4, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*Tanh[x/2])/(2*a*Sqrt[a + a*Cosh[x]])

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((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 4186

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

Rule 4180

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] + (-Dist[(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] + Dist[(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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2531

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] + Dist[(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 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin{align*} \int \frac{x^3}{(a+a \cosh (x))^{3/2}} \, dx &=\frac{\cosh \left (\frac{x}{2}\right ) \int x^3 \text{sech}^3\left (\frac{x}{2}\right ) \, dx}{2 a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}+\frac{\cosh \left (\frac{x}{2}\right ) \int x^3 \text{sech}\left (\frac{x}{2}\right ) \, dx}{4 a \sqrt{a+a \cosh (x)}}-\frac{\left (6 \cosh \left (\frac{x}{2}\right )\right ) \int x \text{sech}\left (\frac{x}{2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}-\frac{\left (3 i \cosh \left (\frac{x}{2}\right )\right ) \int x^2 \log \left (1-i e^{x/2}\right ) \, dx}{2 a \sqrt{a+a \cosh (x)}}+\frac{\left (3 i \cosh \left (\frac{x}{2}\right )\right ) \int x^2 \log \left (1+i e^{x/2}\right ) \, dx}{2 a \sqrt{a+a \cosh (x)}}+\frac{\left (12 i \cosh \left (\frac{x}{2}\right )\right ) \int \log \left (1-i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}-\frac{\left (12 i \cosh \left (\frac{x}{2}\right )\right ) \int \log \left (1+i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}+\frac{\left (6 i \cosh \left (\frac{x}{2}\right )\right ) \int x \text{Li}_2\left (-i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}-\frac{\left (6 i \cosh \left (\frac{x}{2}\right )\right ) \int x \text{Li}_2\left (i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}+\frac{\left (24 i \cosh \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{\left (24 i \cosh \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}-\frac{\left (12 i \cosh \left (\frac{x}{2}\right )\right ) \int \text{Li}_3\left (-i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}+\frac{\left (12 i \cosh \left (\frac{x}{2}\right )\right ) \int \text{Li}_3\left (i e^{x/2}\right ) \, dx}{a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}-\frac{\left (24 i \cosh \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{\left (24 i \cosh \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}\\ &=\frac{3 x^2}{a \sqrt{a+a \cosh (x)}}-\frac{24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{3 i x^2 \cosh \left (\frac{x}{2}\right ) \text{Li}_2\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{12 i x \cosh \left (\frac{x}{2}\right ) \text{Li}_3\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}-\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_4\left (-i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{24 i \cosh \left (\frac{x}{2}\right ) \text{Li}_4\left (i e^{x/2}\right )}{a \sqrt{a+a \cosh (x)}}+\frac{x^3 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a+a \cosh (x)}}\\ \end{align*}

Mathematica [A]  time = 2.92329, size = 716, normalized size = 1.78 \[ -\frac{i \cosh \left (\frac{x}{2}\right ) \left (48 x^2 \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )-48 \left (-x^2-2 i \pi x+\pi ^2+8\right ) \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{-x/2}\right )+96 i \pi x \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )+192 x \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{-x/2}\right )-192 x \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{x/2}\right )+384 \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{-x/2}\right )-48 \pi ^2 \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )+192 i \pi \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{-x/2}\right )-192 i \pi \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i e^{x/2}\right )+384 \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (4,-i e^{-x/2}\right )+384 \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (4,-i e^{x/2}\right )+8 i x^3 \sinh \left (\frac{x}{2}\right )+x^4 \left (-\cosh ^2\left (\frac{x}{2}\right )\right )-4 i \pi x^3 \cosh ^2\left (\frac{x}{2}\right )+6 \pi ^2 x^2 \cosh ^2\left (\frac{x}{2}\right )+48 i x^2 \cosh \left (\frac{x}{2}\right )-8 x^3 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+8 x^3 \log \left (1+i e^{x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )-24 i \pi x^2 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+24 i \pi x^2 \log \left (1-i e^{x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+4 i \pi ^3 x \cosh ^2\left (\frac{x}{2}\right )+7 \pi ^4 \cosh ^2\left (\frac{x}{2}\right )-192 x \log \left (1-i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+24 \pi ^2 x \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+192 x \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )-24 \pi ^2 x \log \left (1-i e^{x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+8 i \pi ^3 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )-8 i \pi ^3 \log \left (1+i e^{x/2}\right ) \cosh ^2\left (\frac{x}{2}\right )+8 i \pi ^3 \cosh ^2\left (\frac{x}{2}\right ) \log \left (\tan \left (\frac{1}{4} (\pi +i x)\right )\right )\right )}{8 (a (\cosh (x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/8)*Cosh[x/2]*((48*I)*x^2*Cosh[x/2] + 7*Pi^4*Cosh[x/2]^2 + (4*I)*Pi^3*x*Cosh[x/2]^2 + 6*Pi^2*x^2*Cosh[x/2]
^2 - (4*I)*Pi*x^3*Cosh[x/2]^2 - x^4*Cosh[x/2]^2 - 192*x*Cosh[x/2]^2*Log[1 - I/E^(x/2)] + (8*I)*Pi^3*Cosh[x/2]^
2*Log[1 + I/E^(x/2)] + 192*x*Cosh[x/2]^2*Log[1 + I/E^(x/2)] + 24*Pi^2*x*Cosh[x/2]^2*Log[1 + I/E^(x/2)] - (24*I
)*Pi*x^2*Cosh[x/2]^2*Log[1 + I/E^(x/2)] - 8*x^3*Cosh[x/2]^2*Log[1 + I/E^(x/2)] - 24*Pi^2*x*Cosh[x/2]^2*Log[1 -
I*E^(x/2)] + (24*I)*Pi*x^2*Cosh[x/2]^2*Log[1 - I*E^(x/2)] - (8*I)*Pi^3*Cosh[x/2]^2*Log[1 + I*E^(x/2)] + 8*x^3
*Cosh[x/2]^2*Log[1 + I*E^(x/2)] + (8*I)*Pi^3*Cosh[x/2]^2*Log[Tan[(Pi + I*x)/4]] - 48*(8 + Pi^2 - (2*I)*Pi*x -
x^2)*Cosh[x/2]^2*PolyLog[2, (-I)/E^(x/2)] + 384*Cosh[x/2]^2*PolyLog[2, I/E^(x/2)] + 48*x^2*Cosh[x/2]^2*PolyLog
[2, (-I)*E^(x/2)] - 48*Pi^2*Cosh[x/2]^2*PolyLog[2, I*E^(x/2)] + (96*I)*Pi*x*Cosh[x/2]^2*PolyLog[2, I*E^(x/2)]
+ (192*I)*Pi*Cosh[x/2]^2*PolyLog[3, (-I)/E^(x/2)] + 192*x*Cosh[x/2]^2*PolyLog[3, (-I)/E^(x/2)] - 192*x*Cosh[x/
2]^2*PolyLog[3, (-I)*E^(x/2)] - (192*I)*Pi*Cosh[x/2]^2*PolyLog[3, I*E^(x/2)] + 384*Cosh[x/2]^2*PolyLog[4, (-I)
/E^(x/2)] + 384*Cosh[x/2]^2*PolyLog[4, (-I)*E^(x/2)] + (8*I)*x^3*Sinh[x/2]))/(a*(1 + Cosh[x]))^(3/2)

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+a\cosh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{8}{27} \, \sqrt{2}{\left (\frac{3 \, e^{\left (\frac{5}{2} \, x\right )} + 8 \, e^{\left (\frac{3}{2} \, x\right )} - 3 \, e^{\left (\frac{1}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}} + \frac{3 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{a^{\frac{3}{2}}}\right )} + 36 \, \sqrt{2} \int \frac{x^{3} e^{\left (\frac{3}{2} \, x\right )}}{9 \,{\left (a^{\frac{3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}\right )}}\,{d x} + 72 \, \sqrt{2} \int \frac{x^{2} e^{\left (\frac{3}{2} \, x\right )}}{9 \,{\left (a^{\frac{3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}\right )}}\,{d x} + 96 \, \sqrt{2} \int \frac{x e^{\left (\frac{3}{2} \, x\right )}}{9 \,{\left (a^{\frac{3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}\right )}}\,{d x} - \frac{4 \,{\left (9 \, \sqrt{2} \sqrt{a} x^{3} + 18 \, \sqrt{2} \sqrt{a} x^{2} + 24 \, \sqrt{2} \sqrt{a} x + 16 \, \sqrt{2} \sqrt{a}\right )} e^{\left (\frac{3}{2} \, x\right )}}{27 \,{\left (a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/27*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^3*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) + 72*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) + 96*sqrt(2)*integrate(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)*sqrt(a)*x^3 + 18*sqrt(2)*sqrt(a)*x^2 + 24*sqrt(2)*sqrt(a)*x + 16*sqrt(2)*sqrt(a))*e^(3/2*x)/(a^2*e^(3*x) +
3*a^2*e^(2*x) + 3*a^2*e^x + a^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cosh \left (x\right ) + a} x^{3}}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a \left (\cosh{\left (x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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