Integrand size = 14, antiderivative size = 386 \[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}} \]
x^3*ln(1+b*exp(2*x)/(2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-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*e xp(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*ex p(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)
Time = 0.13 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.72 \[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\frac {4 x^3 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )-4 x^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+6 x^2 \operatorname {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-6 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-6 x \operatorname {PolyLog}\left (3,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )+6 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+3 \operatorname {PolyLog}\left (4,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}} \]
(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*Poly Log[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])
Time = 1.40 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6162, 3042, 3803, 27, 2694, 27, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3}{a+b \sinh (x) \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 6162 |
| \(\displaystyle \int \frac {x^3}{a+\frac {1}{2} b \sinh (2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {x^3}{a-\frac {1}{2} i b \sin (2 i x)}dx\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle 2 \int -\frac {2 e^{2 x} x^3}{-4 e^{2 x} a-b e^{4 x}+b}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \int \frac {e^{2 x} x^3}{-4 e^{2 x} a-b e^{4 x}+b}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -4 \left (\frac {b \int -\frac {e^{2 x} x^3}{2 \left (2 a+b e^{2 x}-\sqrt {4 a^2+b^2}\right )}dx}{\sqrt {4 a^2+b^2}}-\frac {b \int -\frac {e^{2 x} x^3}{2 \left (2 a+b e^{2 x}+\sqrt {4 a^2+b^2}\right )}dx}{\sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \left (\frac {b \int \frac {e^{2 x} x^3}{2 a+b e^{2 x}+\sqrt {4 a^2+b^2}}dx}{2 \sqrt {4 a^2+b^2}}-\frac {b \int \frac {e^{2 x} x^3}{2 a+b e^{2 x}-\sqrt {4 a^2+b^2}}dx}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a+\sqrt {4 a^2+b^2}}+1\right )dx}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a-\sqrt {4 a^2+b^2}}+1\right )dx}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {3 \left (\int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {3 \left (\int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )de^{2 x}-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )de^{2 x}-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )-\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
-4*(-1/2*(b*((x^3*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/(2*b) - (3*(-1/2*(x^2*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]) + (x*P olyLog[3, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/2 - PolyLog[4, -((b*E ^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/4))/(2*b)))/Sqrt[4*a^2 + b^2] + (b*((x ^3*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/(2*b) - (3*(-1/2*(x^2*P olyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]) + (x*PolyLog[3, -((b* E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/2 - PolyLog[4, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/4))/(2*b)))/(2*Sqrt[4*a^2 + b^2]))
3.9.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
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]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[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]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cosh[(c_.) + (d_.)*(x_)]*(b_.)*Sinh[ (c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> 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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(334)=668\).
Time = 1.55 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(\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 {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 {x^{4}}{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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \operatorname {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 \operatorname {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 \operatorname {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}}}\) | \(687\) |
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+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^3-1/2/(-2*a-(4*a^2+b^2)^(1/2))*x^4-1/(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*p olylog(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))
Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (332) = 664\).
Time = 0.29 (sec) , antiderivative size = 1488, normalized size of antiderivative = 3.85 \[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\text {Too large to display} \]
-(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))*sqr t((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*co sh(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*cos h(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))*sq rt((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^...
Timed out. \[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\text {Timed out} \]
\[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\int { \frac {x^{3}}{b \cosh \left (x\right ) \sinh \left (x\right ) + a} \,d x } \]
\[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\int { \frac {x^{3}}{b \cosh \left (x\right ) \sinh \left (x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx=\int \frac {x^3}{a+b\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]