3.3.57 \(\int \frac {x^3}{a+b \sinh ^2(x)} \, dx\) [257]

Optimal. Leaf size=439 \[ \frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}-\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}} \]

[Out]

1/2*x^3*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^3*ln(1+b*exp(2*x)/(2*a-b+2*a^
(1/2)*(a-b)^(1/2)))/a^(1/2)/(a-b)^(1/2)+3/4*x^2*polylog(2,-b*exp(2*x)/(2*a-b-2*a^(1/2)*(a-b)^(1/2)))/a^(1/2)/(
a-b)^(1/2)-3/4*x^2*polylog(2,-b*exp(2*x)/(2*a-b+2*a^(1/2)*(a-b)^(1/2)))/a^(1/2)/(a-b)^(1/2)-3/4*x*polylog(3,-b
*exp(2*x)/(2*a-b-2*a^(1/2)*(a-b)^(1/2)))/a^(1/2)/(a-b)^(1/2)+3/4*x*polylog(3,-b*exp(2*x)/(2*a-b+2*a^(1/2)*(a-b
)^(1/2)))/a^(1/2)/(a-b)^(1/2)+3/8*polylog(4,-b*exp(2*x)/(2*a-b-2*a^(1/2)*(a-b)^(1/2)))/a^(1/2)/(a-b)^(1/2)-3/8
*polylog(4,-b*exp(2*x)/(2*a-b+2*a^(1/2)*(a-b)^(1/2)))/a^(1/2)/(a-b)^(1/2)

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Rubi [A]
time = 0.55, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5748, 3401, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}+\frac {x^3 \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^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sinh[x]^2),x]

[Out]

(x^3*Log[1 + (b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) - (x^3*Log[1 + (b*E^(2*x)
)/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) + (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a - 2*Sqrt[
a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) - (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b]
- b))])/(4*Sqrt[a]*Sqrt[a - b]) - (3*x*PolyLog[3, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a
]*Sqrt[a - b]) + (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) +
(3*PolyLog[4, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(8*Sqrt[a]*Sqrt[a - b]) - (3*PolyLog[4, -((b*
E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(8*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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {x^3}{a+b \sinh ^2(x)} \, dx &=2 \int \frac {x^3}{2 a-b+b \cosh (2 x)} \, dx\\ &=4 \int \frac {e^{2 x} x^3}{b+2 (2 a-b) e^{2 x}+b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x^3}{-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^3}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a-b}}\\ &=\frac {x^3 \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^3 \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 {3 \int x^2 \log \left (1+\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 {3 \int x^2 \log \left (1+\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^3 \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^3 \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 {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 \int x \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 {3 \int x \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^3 \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^3 \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 {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \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 {3 x \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 {3 \int \text {Li}_3\left (-\frac {2 b e^{2 x}}{-4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right ) \, dx}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 \int \text {Li}_3\left (-\frac {2 b e^{2 x}}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right ) \, dx}{4 \sqrt {a} \sqrt {a-b}}\\ &=\frac {x^3 \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^3 \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 {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \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 {3 x \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 {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{x} \, dx,x,e^{2 x}\right )}{8 \sqrt {a} \sqrt {a-b}}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )}{x} \, dx,x,e^{2 x}\right )}{8 \sqrt {a} \sqrt {a-b}}\\ &=\frac {x^3 \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^3 \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 {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \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 {3 x \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 {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 319, normalized size = 0.73 \begin {gather*} \frac {-4 x^3 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )+4 x^3 \log \left (1-\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )-6 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )+6 x^2 \text {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )+6 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )-6 x \text {PolyLog}\left (3,\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )-3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )+3 \text {PolyLog}\left (4,\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )}{8 \sqrt {a} \sqrt {a-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sinh[x]^2),x]

[Out]

(-4*x^3*Log[1 + (b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b)] + 4*x^3*Log[1 - (b*E^(2*x))/(-2*a + 2*Sqrt[a]*S
qrt[a - b] + b)] - 6*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))] + 6*x^2*PolyLog[2, (b*E^
(2*x))/(-2*a + 2*Sqrt[a]*Sqrt[a - b] + b)] + 6*x*PolyLog[3, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))]
- 6*x*PolyLog[3, (b*E^(2*x))/(-2*a + 2*Sqrt[a]*Sqrt[a - b] + b)] - 3*PolyLog[4, -((b*E^(2*x))/(2*a + 2*Sqrt[a]
*Sqrt[a - b] - b))] + 3*PolyLog[4, (b*E^(2*x))/(-2*a + 2*Sqrt[a]*Sqrt[a - b] + b)])/(8*Sqrt[a]*Sqrt[a - b])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(918\) vs. \(2(351)=702\).
time = 0.86, size = 919, normalized size = 2.09

method	result	size
risch	\(\frac {x^{3} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \sqrt {a \left (a -b \right )}}-\frac {x^{4}}{4 \sqrt {a \left (a -b \right )}}+\frac {3 x^{2} \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \sqrt {a \left (a -b \right )}}-\frac {3 x \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \sqrt {a \left (a -b \right )}}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{8 \sqrt {a \left (a -b \right )}}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) x^{3}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a \,x^{3}}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b \,x^{3}}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {x^{4}}{2 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {a \,x^{4}}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {b \,x^{4}}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) x^{2}}{2 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a \,x^{2}}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b \,x^{2}}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) x}{2 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a x}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b x}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b}{8 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}\)	\(919\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/2/(a*(a-b))^(1/2)*x^3*ln(1-b*exp(2*x)/(2*(a*(a-b))^(1/2)-2*a+b))-1/4/(a*(a-b))^(1/2)*x^4+3/4/(a*(a-b))^(1/2)
*x^2*polylog(2,b*exp(2*x)/(2*(a*(a-b))^(1/2)-2*a+b))-3/4/(a*(a-b))^(1/2)*x*polylog(3,b*exp(2*x)/(2*(a*(a-b))^(
1/2)-2*a+b))+3/8/(a*(a-b))^(1/2)*polylog(4,b*exp(2*x)/(2*(a*(a-b))^(1/2)-2*a+b))+1/(-2*(a*(a-b))^(1/2)-2*a+b)*
ln(1-b*exp(2*x)/(-2*(a*(a-b))^(1/2)-2*a+b))*x^3+1/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*ln(1-b*exp(2*x)/(
-2*(a*(a-b))^(1/2)-2*a+b))*a*x^3-1/2/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*ln(1-b*exp(2*x)/(-2*(a*(a-b))^
(1/2)-2*a+b))*b*x^3-1/2/(-2*(a*(a-b))^(1/2)-2*a+b)*x^4-1/2/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*a*x^4+1/
4/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*b*x^4+3/2/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(2,b*exp(2*x)/(-2*(a*
(a-b))^(1/2)-2*a+b))*x^2+3/2/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(2,b*exp(2*x)/(-2*(a*(a-b))^(1/
2)-2*a+b))*a*x^2-3/4/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(2,b*exp(2*x)/(-2*(a*(a-b))^(1/2)-2*a+b
))*b*x^2-3/2/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(3,b*exp(2*x)/(-2*(a*(a-b))^(1/2)-2*a+b))*x-3/2/(a*(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))*a*x+3/4/(a*(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))*b*x+3/4/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(
4,b*exp(2*x)/(-2*(a*(a-b))^(1/2)-2*a+b))+3/4/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(4,b*exp(2*x)/(
-2*(a*(a-b))^(1/2)-2*a+b))*a-3/8/(a*(a-b))^(1/2)/(-2*(a*(a-b))^(1/2)-2*a+b)*polylog(4,b*exp(2*x)/(-2*(a*(a-b))
^(1/2)-2*a+b))*b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*sinh(x)^2),x, algorithm="maxima")

[Out]

integrate(x^3/(b*sinh(x)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1655 vs. \(2 (339) = 678\).
time = 0.42, size = 1655, normalized size = 3.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*sinh(x)^2),x, algorithm="fricas")

[Out]

-1/2*(b*x^3*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^3*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^3*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^
3*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) + 3*b*x^2*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) + 3*b*x^2*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)
- 3*b*x^2*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))*sq
rt((a^2 - a*b)/b^2))*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b) + b)/b + 1) - 3*b*x^2*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*sq
rt((a^2 - a*b)/b^2) - 2*a + b)/b) - b)/b + 1) - 6*b*x*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) - 6*b*x*sqrt((a^2 - a*b)/b^2)*polylog(3, -((2*a - b)*cosh(x) + (2*a - b)*sinh(x) - 2*(b*cosh(x) + b*sin
h(x))*sqrt((a^2 - a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 - a*b)/b^2) + 2*a - b)/b)/b) + 6*b*x*sqrt((a^2 - a*b)/b^2)*p
olylog(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) + 6*b*x*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)
+ 6*b*sqrt((a^2 - a*b)/b^2)*polylog(4, ((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) + 6*b*sqrt((a^2 - a*b)/b^2)*polylog(4, -(
(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) - 6*b*sqrt((a^2 - a*b)/b^2)*polylog(4, ((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) - 6*b*sqrt((a
^2 - a*b)/b^2)*polylog(4, -((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{a + b \sinh ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*sinh(x)**2),x)

[Out]

Integral(x**3/(a + b*sinh(x)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*sinh(x)^2),x, algorithm="giac")

[Out]

integrate(x^3/(b*sinh(x)^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*sinh(x)^2),x)

[Out]

int(x^3/(a + b*sinh(x)^2), x)

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