3.140 \(\int \frac{x^2}{\sqrt{a+a \cosh (c+d x)}} \, dx\)

Optimal. Leaf size=269 \[ -\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (2,-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a \cosh (c+d x)+a}}+\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (2,i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a \cosh (c+d x)+a}}+\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (3,-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a \cosh (c+d x)+a}}-\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (3,i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a \cosh (c+d x)+a}}+\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a \cosh (c+d x)+a}} \]

[Out]

(4*x^2*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((8*I)*x*Cosh[c/2 + (d*x
)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((8*I)*x*Cosh[c/2 + (d*x)/2]*PolyLo
g[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^(c/
2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (d*x)/2)])/
(d^3*Sqrt[a + a*Cosh[c + d*x]])

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Rubi [A]  time = 0.163233, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 4180, 2531, 2282, 6589} \[ -\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (2,-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a \cosh (c+d x)+a}}+\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (2,i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a \cosh (c+d x)+a}}+\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (3,-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a \cosh (c+d x)+a}}-\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{PolyLog}\left (3,i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a \cosh (c+d x)+a}}+\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a \cosh (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

(4*x^2*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((8*I)*x*Cosh[c/2 + (d*x
)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((8*I)*x*Cosh[c/2 + (d*x)/2]*PolyLo
g[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^(c/
2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (d*x)/2)])/
(d^3*Sqrt[a + a*Cosh[c + d*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 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}{\sqrt{a+a \cosh (c+d x)}} \, dx &=\frac{\sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right ) \int x^2 \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right ) \, dx}{\sqrt{a+a \cosh (c+d x)}}\\ &=\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cosh (c+d x)}}-\frac{\left (4 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x \log \left (1-i e^{\frac{c}{2}+\frac{d x}{2}}\right ) \, dx}{d \sqrt{a+a \cosh (c+d x)}}+\frac{\left (4 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x \log \left (1+i e^{\frac{c}{2}+\frac{d x}{2}}\right ) \, dx}{d \sqrt{a+a \cosh (c+d x)}}\\ &=\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cosh (c+d x)}}-\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{\left (8 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int \text{Li}_2\left (-i e^{\frac{c}{2}+\frac{d x}{2}}\right ) \, dx}{d^2 \sqrt{a+a \cosh (c+d x)}}-\frac{\left (8 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int \text{Li}_2\left (i e^{\frac{c}{2}+\frac{d x}{2}}\right ) \, dx}{d^2 \sqrt{a+a \cosh (c+d x)}}\\ &=\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cosh (c+d x)}}-\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{\left (16 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a+a \cosh (c+d x)}}-\frac{\left (16 i \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a+a \cosh (c+d x)}}\\ &=\frac{4 x^2 \tan ^{-1}\left (e^{\frac{c}{2}+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cosh (c+d x)}}-\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{8 i x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^2 \sqrt{a+a \cosh (c+d x)}}+\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_3\left (-i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a+a \cosh (c+d x)}}-\frac{16 i \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_3\left (i e^{\frac{c}{2}+\frac{d x}{2}}\right )}{d^3 \sqrt{a+a \cosh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 1.70702, size = 163, normalized size = 0.61 \[ \frac{2 i \cosh \left (\frac{1}{2} (c+d x)\right ) \left (-4 d x \text{PolyLog}\left (2,-i e^{\frac{1}{2} (c+d x)}\right )+4 d x \text{PolyLog}\left (2,i e^{\frac{1}{2} (c+d x)}\right )+8 \text{PolyLog}\left (3,-i e^{\frac{1}{2} (c+d x)}\right )-8 \text{PolyLog}\left (3,i e^{\frac{1}{2} (c+d x)}\right )+d^2 x^2 \log \left (1-i e^{\frac{1}{2} (c+d x)}\right )-d^2 x^2 \log \left (1+i e^{\frac{1}{2} (c+d x)}\right )\right )}{d^3 \sqrt{a (\cosh (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

((2*I)*Cosh[(c + d*x)/2]*(d^2*x^2*Log[1 - I*E^((c + d*x)/2)] - d^2*x^2*Log[1 + I*E^((c + d*x)/2)] - 4*d*x*Poly
Log[2, (-I)*E^((c + d*x)/2)] + 4*d*x*PolyLog[2, I*E^((c + d*x)/2)] + 8*PolyLog[3, (-I)*E^((c + d*x)/2)] - 8*Po
lyLog[3, I*E^((c + d*x)/2)]))/(d^3*Sqrt[a*(1 + Cosh[c + d*x])])

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt{a+a\cosh \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+a*cosh(d*x+c))^(1/2),x)

[Out]

int(x^2/(a+a*cosh(d*x+c))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, \sqrt{2} d^{2} \int \frac{x^{2} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{\sqrt{a} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, \sqrt{a} d^{2} e^{\left (d x + c\right )} + \sqrt{a} d^{2}}\,{d x} + 8 \, \sqrt{2} d \int \frac{x e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{\sqrt{a} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, \sqrt{a} d^{2} e^{\left (d x + c\right )} + \sqrt{a} d^{2}}\,{d x} + 16 \, \sqrt{2}{\left (\frac{e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{{\left (\sqrt{a} d^{2} e^{\left (d x + c\right )} + \sqrt{a} d^{2}\right )} d} + \frac{\arctan \left (e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}\right )}{\sqrt{a} d^{3}}\right )} - \frac{2 \,{\left (\sqrt{2} d^{2} x^{2} e^{\left (\frac{1}{2} \, c\right )} + 4 \, \sqrt{2} d x e^{\left (\frac{1}{2} \, c\right )} + 8 \, \sqrt{2} e^{\left (\frac{1}{2} \, c\right )}\right )} e^{\left (\frac{1}{2} \, d x\right )}}{\sqrt{a} d^{3} e^{\left (d x + c\right )} + \sqrt{a} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(2)*d^2*integrate(x^2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^2*e^(2*d*x + 2*c) + 2*sqrt(a)*d^2*e^(d*x + c) + sqr
t(a)*d^2), x) + 8*sqrt(2)*d*integrate(x*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^2*e^(2*d*x + 2*c) + 2*sqrt(a)*d^2*e^(d*
x + c) + sqrt(a)*d^2), x) + 16*sqrt(2)*(e^(1/2*d*x + 1/2*c)/((sqrt(a)*d^2*e^(d*x + c) + sqrt(a)*d^2)*d) + arct
an(e^(1/2*d*x + 1/2*c))/(sqrt(a)*d^3)) - 2*(sqrt(2)*d^2*x^2*e^(1/2*c) + 4*sqrt(2)*d*x*e^(1/2*c) + 8*sqrt(2)*e^
(1/2*c))*e^(1/2*d*x)/(sqrt(a)*d^3*e^(d*x + c) + sqrt(a)*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(x**2/sqrt(a*(cosh(c + d*x) + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(a*cosh(d*x + c) + a), x)