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

Optimal. Leaf size=248 \[ -\frac{2 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{2 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{4 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{4 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^2 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}}+\frac{2 x}{a \sqrt{a \cosh (x)+a}}-\frac{4 \cosh \left (\frac{x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )\right )}{a \sqrt{a \cosh (x)+a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.191267, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3319, 4186, 3770, 4180, 2531, 2282, 6589} \[ -\frac{2 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{2 i x \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{4 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}-\frac{4 i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x^2 \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}}+\frac{2 x}{a \sqrt{a \cosh (x)+a}}-\frac{4 \cosh \left (\frac{x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )\right )}{a \sqrt{a \cosh (x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + a*Cosh[x])^(3/2),x]

[Out]

(2*x)/(a*Sqrt[a + a*Cosh[x]]) + (x^2*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - (4*ArcTan[Sinh[x/2]]
*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - ((2*I)*x*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (
(2*I)*x*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((4*I)*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/
(a*Sqrt[a + a*Cosh[x]]) - ((4*I)*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^2*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 3770

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

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

Mathematica [A]  time = 0.879235, size = 214, normalized size = 0.86 \[ \frac{\cosh \left (\frac{x}{2}\right ) \left (-4 i x \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i \left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )\right )+4 i x \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i \left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )\right )+8 i \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,-i \left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )\right )-8 i \cosh ^2\left (\frac{x}{2}\right ) \text{PolyLog}\left (3,i \left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )\right )+x^2 \sinh \left (\frac{x}{2}\right )+2 x^2 \cosh ^2\left (\frac{x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )+4 x \cosh \left (\frac{x}{2}\right )-16 \cosh ^2\left (\frac{x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )\right )\right )}{(a (\cosh (x)+1))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cosh[x/2]*(4*x*Cosh[x/2] - 16*ArcTan[Cosh[x/2] + Sinh[x/2]]*Cosh[x/2]^2 + 2*x^2*ArcTan[Cosh[x/2] + Sinh[x/2]]
*Cosh[x/2]^2 - (4*I)*x*Cosh[x/2]^2*PolyLog[2, (-I)*(Cosh[x/2] + Sinh[x/2])] + (4*I)*x*Cosh[x/2]^2*PolyLog[2, I
*(Cosh[x/2] + Sinh[x/2])] + (8*I)*Cosh[x/2]^2*PolyLog[3, (-I)*(Cosh[x/2] + Sinh[x/2])] - (8*I)*Cosh[x/2]^2*Pol
yLog[3, I*(Cosh[x/2] + Sinh[x/2])] + x^2*Sinh[x/2]))/(a*(1 + Cosh[x]))^(3/2)

________________________________________________________________________________________

Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^2/(a+a*cosh(x))^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4}{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^{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} + 48 \, \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} x^{2} + 12 \, \sqrt{2} x + 8 \, \sqrt{2}\right )} e^{\left (\frac{3}{2} \, x\right )}}{27 \,{\left (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}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*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)*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))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\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^2/(a+a*cosh(x))^(3/2),x, algorithm="giac")

[Out]

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