3.211 \(\int \frac{x^2}{a+b \cosh ^2(x)} \, dx\)
Optimal. Leaf size=291 \[ \frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{x^2 \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}+1\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x^2 \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}+1\right )}{2 \sqrt{a} \sqrt{a+b}} \]
[Out]
(x^2*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x^2*Log[1 + (b*E^(2*x)
)/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[
a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/
(2*Sqrt[a]*Sqrt[a + b]) - PolyLog[3, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])
+ PolyLog[3, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])
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Rubi [A] time = 0.566805, antiderivative size = 291, 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 =
{5630, 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*Cosh[x]^2),x]
[Out]
(x^2*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x^2*Log[1 + (b*E^(2*x)
)/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[
a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/
(2*Sqrt[a]*Sqrt[a + b]) - PolyLog[3, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])
+ PolyLog[3, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])
Rule 5630
Int[(Cosh[(c_.) + (d_.)*(x_)]^2*(b_.) + (a_))^(n_)*(x_)^(m_.), 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 \cosh ^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+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+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+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+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+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{2 a+b-2 \sqrt{a} \sqrt{a+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+b+2 \sqrt{a} \sqrt{a+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+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x^2 \log \left (1+\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (-\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{Li}_3\left (-\frac{b e^{2 x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.717135, size = 221, normalized size = 0.76 \[ \frac{2 x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )-2 x \text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )-\text{PolyLog}\left (3,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )+\text{PolyLog}\left (3,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )+2 x^2 \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}+1\right )-2 x^2 \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}+1\right )}{4 \sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(a + b*Cosh[x]^2),x]
[Out]
(2*x^2*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 2*x^2*Log[1 + (b*E^(2*x))/(2*a + b + 2*Sqrt[a]
*Sqrt[a + b])] + 2*x*PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] - 2*x*PolyLog[2, -((b*E^(2*x
))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] - PolyLog[3, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + PolyLo
g[3, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b])
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Maple [B] time = 0.038, size = 686, normalized size = 2.4 \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*cosh(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 \cosh \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*cosh(x)^2),x, algorithm="maxima")
[Out]
integrate(x^2/(b*cosh(x)^2 + a), x)
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Fricas [C] time = 2.12077, size = 2967, normalized size = 10.2 \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)^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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(a+b*cosh(x)**2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b \cosh \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*cosh(x)^2),x, algorithm="giac")
[Out]
integrate(x^2/(b*cosh(x)^2 + a), x)