3.867 \(\int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx\)

Optimal. Leaf size=386 \[ \frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}+\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{\sqrt {4 a^2+b^2}}-\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \]

[Out]

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

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Rubi [A]  time = 0.60, antiderivative size = 386, 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 = {5628, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}\right )}{4 \sqrt {4 a^2+b^2}}+\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{\sqrt {4 a^2+b^2}}-\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Cosh[x]*Sinh[x]),x]

[Out]

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

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 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 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 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 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 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 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]

Rule 6609

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

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Mathematica [A]  time = 0.41, size = 279, normalized size = 0.72 \[ \frac {6 x^2 \text {Li}_2\left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}-2 a}\right )-6 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-6 x \text {Li}_3\left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}-2 a}\right )+6 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+3 \text {Li}_4\left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}-2 a}\right )-3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+4 x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )-4 x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{4 \sqrt {4 a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Cosh[x]*Sinh[x]),x]

[Out]

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

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fricas [C]  time = 0.54, size = 1488, normalized size = 3.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x^3*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^3*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^3*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^3*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) + 3*b*x^2*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) + 3*b*x^2*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) - 3*b*x^2*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*s
inh(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) - 3*b*x^2*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) - 6*b*x*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) - 6*b*x*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))*s
qrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b)/b) + 6*b*x*sqrt((4*a^2 + b^2)/b^2)*polylog(3, (2*a*cosh(x) + 2*a*sin
h(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) + 6*b*x*s
qrt((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) + 6*b*sqrt((4*a^2 + b^2)/b^2)*polylog(4, (2*a*cosh(x) + 2*a*si
nh(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) + 6*b*s
qrt((4*a^2 + b^2)/b^2)*polylog(4, -(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) - 6*b*sqrt((4*a^2 + b^2)/b^2)*polylog(4, (2*a*cosh(x) + 2*a*s
inh(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) - 6*b*s
qrt((4*a^2 + b^2)/b^2)*polylog(4, -(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \cosh \relax (x) \sinh \relax (x) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.32, size = 687, normalized size = 1.78 \[ \frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x^{3}}{-2 a -\sqrt {4 a^{2}+b^{2}}}-\frac {x^{4}}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a \,x^{3}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {a \,x^{4}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x^{2}}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a \,x^{2}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a x}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{4 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a}{2 \sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {x^{3} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}-\frac {x^{4}}{2 \sqrt {4 a^{2}+b^{2}}}+\frac {3 x^{2} \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}-\frac {3 x \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{4 \sqrt {4 a^{2}+b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*cosh(x)*sinh(x)),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \cosh \relax (x) \sinh \relax (x) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{a+b\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*cosh(x)*sinh(x)),x)

[Out]

int(x^3/(a + b*cosh(x)*sinh(x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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