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

Optimal. Leaf size=327 \[ \frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{x^2 \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}} \]

[Out]

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

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Rubi [A]  time = 0.538426, antiderivative size = 327, 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 = {5629, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{x^2 \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sinh[x]^2),x]

[Out]

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

Rule 5629

Int[(x_)^(m_.)*((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]^2)^(n_), x_Symbol] :> Dist[1/2^n, Int[x^m*(2*a - b + b*C
osh[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n,
-1] || (EqQ[m, 1] && EqQ[n, -2]))

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && 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 \sinh ^2(x)} \, dx &=2 \int \frac{x^2}{2 a-b+b \cosh (2 x)} \, dx\\ &=4 \int \frac{e^{2 x} x^2}{b+2 (2 a-b) e^{2 x}+b e^{4 x}} \, dx\\ &=\frac{(2 b) \int \frac{e^{2 x} x^2}{-4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)+2 b e^{2 x}} \, dx}{\sqrt{a} \sqrt{a-b}}-\frac{(2 b) \int \frac{e^{2 x} x^2}{4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)+2 b e^{2 x}} \, dx}{\sqrt{a} \sqrt{a-b}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{\int x \log \left (1+\frac{2 b e^{2 x}}{-4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)}\right ) \, dx}{\sqrt{a} \sqrt{a-b}}+\frac{\int x \log \left (1+\frac{2 b e^{2 x}}{4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)}\right ) \, dx}{\sqrt{a} \sqrt{a-b}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{\int \text{Li}_2\left (-\frac{2 b e^{2 x}}{-4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a-b}}+\frac{\int \text{Li}_2\left (-\frac{2 b e^{2 x}}{4 \sqrt{a} \sqrt{a-b}+2 (2 a-b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a-b}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt{a} \sqrt{a-b}}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-2 a+2 \sqrt{a} \sqrt{a-b}+b}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt{a} \sqrt{a-b}}\\ &=\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a-2 \sqrt{a} \sqrt{a-b}-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{4 \sqrt{a} \sqrt{a-b}}\\ \end{align*}

Mathematica [A]  time = 0.693783, size = 240, normalized size = 0.73 \[ \frac{-2 x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )+2 x \text{PolyLog}\left (2,\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}-2 a+b}\right )+\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )-\text{PolyLog}\left (3,\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}-2 a+b}\right )-2 x^2 \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )+2 x^2 \log \left (1-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}-2 a+b}\right )}{4 \sqrt{a} \sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Sinh[x]^2),x]

[Out]

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

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Maple [B]  time = 0.049, size = 710, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(b*sinh(x)^2 + a), x)

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Fricas [C]  time = 2.37248, size = 2967, normalized size = 9.07 \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*sinh(x)^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(a + b*sinh(x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(b*sinh(x)^2 + a), x)