\(\int \frac {x^2}{a+b \sinh ^2(x)} \, dx\) [258]

Optimal result

Integrand size = 14, antiderivative size = 327 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5748, 3401, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]

[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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 3401

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)/(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))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5748

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 6724

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*} \text {integral}& = 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 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\int \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\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 {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (3,-\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] (verified)

Time = 0.34 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {-x^2 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-x^2 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+x^2 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+x^2 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-2 x \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-2 x \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+2 x \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+2 x \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+2 \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )}{2 \sqrt {a (a-b)}} \]

[In]

Integrate[x^2/(a + b*Sinh[x]^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs. \(2(261)=522\).

Time = 0.13 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {2 x^{3}}{3 \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) x^{2}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) x}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {2 x^{3} a}{3 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a \,x^{2}}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a x}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {b \,x^{3}}{3 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b \,x^{2}}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b x}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b}{4 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {x^{3}}{3 \sqrt {\left (a -b \right ) a}}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \sqrt {\left (a -b \right ) a}}+\frac {x \operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \sqrt {\left (a -b \right ) a}}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{4 \sqrt {\left (a -b \right ) a}}\)	\(710\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (252) = 504\).

Time = 0.30 (sec) , antiderivative size = 1247, normalized size of antiderivative = 3.81 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\text {Too large to display} \]

[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)

Sympy [F]

\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int \frac {x^{2}}{a + b \sinh ^{2}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(x**2/(a + b*sinh(x)**2), x)

Maxima [F]

\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{2}}{b \sinh \left (x\right )^{2} + a} \,d x } \]

[In]

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

[Out]

integrate(x^2/(b*sinh(x)^2 + a), x)

Giac [F]

\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{2}}{b \sinh \left (x\right )^{2} + a} \,d x } \]

[In]

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

[Out]

integrate(x^2/(b*sinh(x)^2 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int \frac {x^2}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \]

[In]

int(x^2/(a + b*sinh(x)^2),x)

[Out]

int(x^2/(a + b*sinh(x)^2), x)