\(\int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx\) [144]

Optimal result

Integrand size = 14, antiderivative size = 402 \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]

[Out]

3*x^2/a/(a+a*cosh(x))^(1/2)-24*x*arctan(exp(1/2*x))*cosh(1/2*x)/a/(a+a*cosh(x))^(1/2)+x^3*arctan(exp(1/2*x))*c
osh(1/2*x)/a/(a+a*cosh(x))^(1/2)+24*I*cosh(1/2*x)*polylog(2,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)-3*I*x^2*cosh(
1/2*x)*polylog(2,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)-24*I*cosh(1/2*x)*polylog(2,I*exp(1/2*x))/a/(a+a*cosh(x))
^(1/2)+3*I*x^2*cosh(1/2*x)*polylog(2,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+12*I*x*cosh(1/2*x)*polylog(3,-I*exp(1
/2*x))/a/(a+a*cosh(x))^(1/2)-12*I*x*cosh(1/2*x)*polylog(3,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)-24*I*cosh(1/2*x)
*polylog(4,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+24*I*cosh(1/2*x)*polylog(4,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)
+1/2*x^3*tanh(1/2*x)/a/(a+a*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3400, 4271, 4265, 2317, 2438, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {12 i x \operatorname {PolyLog}\left (3,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {12 i x \operatorname {PolyLog}\left (3,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \operatorname {PolyLog}\left (4,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \operatorname {PolyLog}\left (4,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {3 x^2}{a \sqrt {a \cosh (x)+a}} \]

[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 2317

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 2320

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 2438

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 2611

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 3400

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 4265

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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 6724

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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = \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 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \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 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \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 ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,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 \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (24 i \cosh \left (\frac {x}{2}\right )\right ) \text {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 ) \text {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 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \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 ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,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 \operatorname {PolyLog}\left (3,-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,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 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \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 ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \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 ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,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] (verified)

Time = 0.68 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.58 \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (6 x^2 \cosh \left (\frac {x}{2}\right )+8 i \cosh ^2\left (\frac {x}{2}\right ) \left (-3 x \log \left (1-i e^{x/2}\right )+\frac {1}{8} x^3 \log \left (1-i e^{x/2}\right )+3 x \log \left (1+i e^{x/2}\right )-\frac {1}{8} x^3 \log \left (1+i e^{x/2}\right )-\frac {3}{4} \left (-8+x^2\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+\frac {3}{4} \left (-8+x^2\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )+3 x \operatorname {PolyLog}\left (3,-i e^{x/2}\right )-3 x \operatorname {PolyLog}\left (3,i e^{x/2}\right )-6 \operatorname {PolyLog}\left (4,-i e^{x/2}\right )+6 \operatorname {PolyLog}\left (4,i e^{x/2}\right )\right )+x^3 \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \]

[In]

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

[Out]

(Cosh[x/2]*(6*x^2*Cosh[x/2] + (8*I)*Cosh[x/2]^2*(-3*x*Log[1 - I*E^(x/2)] + (x^3*Log[1 - I*E^(x/2)])/8 + 3*x*Lo
g[1 + I*E^(x/2)] - (x^3*Log[1 + I*E^(x/2)])/8 - (3*(-8 + x^2)*PolyLog[2, (-I)*E^(x/2)])/4 + (3*(-8 + x^2)*Poly
Log[2, I*E^(x/2)])/4 + 3*x*PolyLog[3, (-I)*E^(x/2)] - 3*x*PolyLog[3, I*E^(x/2)] - 6*PolyLog[4, (-I)*E^(x/2)] +
6*PolyLog[4, I*E^(x/2)]) + x^3*Sinh[x/2]))/(a*(1 + Cosh[x]))^(3/2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^3/(a + a*cosh(x))^(3/2),x)

[Out]

int(x^3/(a + a*cosh(x))^(3/2), x)