3.868 \(\int \frac{x^2}{a+b \cosh (x) \sinh (x)} \, dx\)

Optimal. Leaf size=281 \[ \frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )}{\sqrt{4 a^2+b^2}}-\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{x^2 \log \left (\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}+1\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}+1\right )}{\sqrt{4 a^2+b^2}} \]

[Out]

(x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[
4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (x*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2]
- (x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - PolyLog[3, -((b*E^(2*x))/(2*a
- Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*
a^2 + b^2])

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Rubi [A]  time = 0.515022, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5628, 3322, 2264, 2190, 2531, 2282, 6589} \[ \frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )}{\sqrt{4 a^2+b^2}}-\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{x^2 \log \left (\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}+1\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}+1\right )}{\sqrt{4 a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Cosh[x]*Sinh[x]),x]

[Out]

(x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[
4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (x*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2]
- (x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - PolyLog[3, -((b*E^(2*x))/(2*a
- Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*
a^2 + b^2])

Rule 5628

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cosh[(c_.) + (d_.)*(x_)]*(b_.)*Sinh[(c_.) + (d_.)*(x_)])^(n_.), x_Symbo
l] :> Int[(e + f*x)^m*(a + (b*Sinh[2*c + 2*d*x])/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && 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+b \cosh (x) \sinh (x)} \, dx &=\int \frac{x^2}{a+\frac{1}{2} b \sinh (2 x)} \, dx\\ &=2 \int \frac{e^{2 x} x^2}{-\frac{b}{2}+2 a e^{2 x}+\frac{1}{2} b e^{4 x}} \, dx\\ &=\frac{(2 b) \int \frac{e^{2 x} x^2}{2 a-\sqrt{4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt{4 a^2+b^2}}-\frac{(2 b) \int \frac{e^{2 x} x^2}{2 a+\sqrt{4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt{4 a^2+b^2}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{2 \int x \log \left (1+\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right ) \, dx}{\sqrt{4 a^2+b^2}}+\frac{2 \int x \log \left (1+\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right ) \, dx}{\sqrt{4 a^2+b^2}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{\int \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right ) \, dx}{\sqrt{4 a^2+b^2}}+\frac{\int \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right ) \, dx}{\sqrt{4 a^2+b^2}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-2 a+\sqrt{4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{2 a+\sqrt{4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt{4 a^2+b^2}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2}}-\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}\right )}{2 \sqrt{4 a^2+b^2}}+\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a+\sqrt{4 a^2+b^2}}\right )}{2 \sqrt{4 a^2+b^2}}\\ \end{align*}

Mathematica [A]  time = 0.25076, size = 210, normalized size = 0.75 \[ \frac{2 x \text{PolyLog}\left (2,\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}-2 a}\right )-2 x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )-\text{PolyLog}\left (3,\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}-2 a}\right )+\text{PolyLog}\left (3,-\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}\right )+2 x^2 \log \left (\frac{b e^{2 x}}{2 a-\sqrt{4 a^2+b^2}}+1\right )-2 x^2 \log \left (\frac{b e^{2 x}}{\sqrt{4 a^2+b^2}+2 a}+1\right )}{2 \sqrt{4 a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])] - 2*x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])] +
2*x*PolyLog[2, (b*E^(2*x))/(-2*a + Sqrt[4*a^2 + b^2])] - 2*x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2
]))] - PolyLog[3, (b*E^(2*x))/(-2*a + Sqrt[4*a^2 + b^2])] + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])
)])/(2*Sqrt[4*a^2 + b^2])

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Maple [B]  time = 0.053, size = 530, normalized size = 1.9 \begin{align*} -{\frac{2\,{x}^{3}}{3} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}+{{x}^{2}\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}+{x{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}-{\frac{1}{2}{\it polylog} \left ( 3,{b{{\rm e}^{2\,x}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}-{\frac{4\,a{x}^{3}}{3}{\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}+2\,{\frac{a{x}^{2}}{\sqrt{4\,{a}^{2}+{b}^{2}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) }\ln \left ( 1-{\frac{b{{\rm e}^{2\,x}}}{-2\,a-\sqrt{4\,{a}^{2}+{b}^{2}}}} \right ) }+2\,{\frac{ax}{\sqrt{4\,{a}^{2}+{b}^{2}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) }{\it polylog} \left ( 2,{\frac{b{{\rm e}^{2\,x}}}{-2\,a-\sqrt{4\,{a}^{2}+{b}^{2}}}} \right ) }-{a{\it polylog} \left ( 3,{b{{\rm e}^{2\,x}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}} \left ( -2\,a-\sqrt{4\,{a}^{2}+{b}^{2}} \right ) ^{-1}}-{\frac{2\,{x}^{3}}{3}{\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}}}+{{x}^{2}\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( \sqrt{4\,{a}^{2}+{b}^{2}}-2\,a \right ) ^{-1}} \right ){\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}}}+{x{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( \sqrt{4\,{a}^{2}+{b}^{2}}-2\,a \right ) ^{-1}} \right ){\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}}}-{\frac{1}{2}{\it polylog} \left ( 3,{b{{\rm e}^{2\,x}} \left ( \sqrt{4\,{a}^{2}+{b}^{2}}-2\,a \right ) ^{-1}} \right ){\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(-2*a-(4*a^2+b^2)^(1/2))*x^3+1/(-2*a-(4*a^2+b^2)^(1/2))*x^2*ln(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))+1/(
-2*a-(4*a^2+b^2)^(1/2))*x*polylog(2,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-1/2/(-2*a-(4*a^2+b^2)^(1/2))*polylog(
3,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-4/3/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*a*x^3+2/(4*a^2+b^2)^(1/2
)/(-2*a-(4*a^2+b^2)^(1/2))*a*x^2*ln(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))+2/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^
2)^(1/2))*a*x*polylog(2,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-1/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*a*po
lylog(3,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-2/3/(4*a^2+b^2)^(1/2)*x^3+1/(4*a^2+b^2)^(1/2)*x^2*ln(1-b*exp(2*x)
/((4*a^2+b^2)^(1/2)-2*a))+1/(4*a^2+b^2)^(1/2)*x*polylog(2,b*exp(2*x)/((4*a^2+b^2)^(1/2)-2*a))-1/2/(4*a^2+b^2)^
(1/2)*polylog(3,b*exp(2*x)/((4*a^2+b^2)^(1/2)-2*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.98174, size = 2738, normalized size = 9.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b
^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b) + b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(-((2*a*cosh(x) + 2
*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) - b)/
b) - b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2
)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b) - b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(-((2*a*cosh(x) +
2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) - b)
/b) + 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 +
b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b + 1) + 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(((2*a
*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2
*a)/b) - b)/b + 1) - 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x)
)*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b + 1) - 2*b*x*sqrt((4*a^2 + b^2)/b^
2)*dilog(((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 +
b^2)/b^2) - 2*a)/b) - b)/b + 1) - 2*b*sqrt((4*a^2 + b^2)/b^2)*polylog(3, (2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(
x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b)/b) - 2*b*sqrt((4*a^2 + b^2
)/b^2)*polylog(3, -(2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt
((4*a^2 + b^2)/b^2) + 2*a)/b)/b) + 2*b*sqrt((4*a^2 + b^2)/b^2)*polylog(3, (2*a*cosh(x) + 2*a*sinh(x) + (b*cosh
(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b)/b) + 2*b*sqrt((4*a^2 + b^2
)/b^2)*polylog(3, -(2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt(
(4*a^2 + b^2)/b^2) - 2*a)/b)/b))/(4*a^2 + b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*cosh(x)*sinh(x)),x)

[Out]

Integral(x**2/(a + b*sinh(x)*cosh(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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