Integrand size = 14, antiderivative size = 439 \[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{8 \sqrt {a} \sqrt {a-b}} \]
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)
Time = 0.51 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.36 \[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\frac {-x^3 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-x^3 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+x^3 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+x^3 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-3 x^2 \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-3 x^2 \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+3 x^2 \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+3 x^2 \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+6 x \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+6 x \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-6 x \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-6 x \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-6 \operatorname {PolyLog}\left (4,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )}{2 \sqrt {a (a-b)}} \]
(-(x^3*Log[1 - E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b]]) - x^3*Log[1 + E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b]] + x^3*Log[1 - E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b)] + b)/b]] + x^3*Log[1 + E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b )] + b)/b]] - 3*x^2*PolyLog[2, -(E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b ])] - 3*x^2*PolyLog[2, E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b]] + 3*x^2 *PolyLog[2, -(E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b)] + b)/b])] + 3*x^2*PolyLog [2, E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b)] + b)/b]] + 6*x*PolyLog[3, -(E^x/Sqr t[(-2*a - 2*Sqrt[a*(a - b)] + b)/b])] + 6*x*PolyLog[3, E^x/Sqrt[(-2*a - 2* Sqrt[a*(a - b)] + b)/b]] - 6*x*PolyLog[3, -(E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b)] + b)/b])] - 6*x*PolyLog[3, E^x/Sqrt[(-2*a + 2*Sqrt[a*(a - b)] + b)/b] ] - 6*PolyLog[4, -(E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b])] - 6*PolyLo g[4, E^x/Sqrt[(-2*a - 2*Sqrt[a*(a - b)] + b)/b]] + 6*PolyLog[4, -(E^x/Sqrt [(-2*a + 2*Sqrt[a*(a - b)] + b)/b])] + 6*PolyLog[4, E^x/Sqrt[(-2*a + 2*Sqr t[a*(a - b)] + b)/b]])/(2*Sqrt[a*(a - b)])
Time = 1.47 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6163, 3042, 3801, 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.
| \(\displaystyle \int \frac {x^3}{a+b \sinh ^2(x)} \, dx\) |
| \(\Big \downarrow \) 6163 |
| \(\displaystyle 2 \int \frac {x^3}{2 a-b+b \cosh (2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \int \frac {x^3}{2 a-b+b \sin \left (2 i x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3801 |
| \(\displaystyle 4 \int \frac {e^{2 x} x^3}{2 e^{2 x} (2 a-b)+b e^{4 x}+b}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle 4 \left (\frac {b \int \frac {e^{2 x} x^3}{2 \left (2 a-2 \sqrt {a-b} \sqrt {a}+b e^{2 x}-b\right )}dx}{2 \sqrt {a} \sqrt {a-b}}-\frac {b \int \frac {e^{2 x} x^3}{2 \left (2 a+2 \sqrt {a-b} \sqrt {a}+b e^{2 x}-b\right )}dx}{2 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {b \int \frac {e^{2 x} x^3}{2 a-2 \sqrt {a-b} \sqrt {a}+b e^{2 x}-b}dx}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \int \frac {e^{2 x} x^3}{2 a+2 \sqrt {a-b} \sqrt {a}+b e^{2 x}-b}dx}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a-2 \sqrt {a-b} \sqrt {a}-b}+1\right )dx}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a+2 \sqrt {a-b} \sqrt {a}-b}+1\right )dx}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (\int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (\int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )de^{2 x}-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )de^{2 x}-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )-\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 b}-\frac {3 \left (-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )-\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a-b}}\right )\) |
4*((b*((x^3*Log[1 + (b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*b) - (3*(-1/2*(x^2*PolyLog[2, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b) )]) + (x*PolyLog[3, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/2 - PolyLog[4, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))]/4))/(2*b)))/ (4*Sqrt[a]*Sqrt[a - b]) - (b*((x^3*Log[1 + (b*E^(2*x))/(2*a + 2*Sqrt[a]*Sq rt[a - b] - b)])/(2*b) - (3*(-1/2*(x^2*PolyLog[2, -((b*E^(2*x))/(2*a + 2*S qrt[a]*Sqrt[a - b] - b))]) + (x*PolyLog[3, -((b*E^(2*x))/(2*a + 2*Sqrt[a]* Sqrt[a - b] - b))])/2 - PolyLog[4, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))]/4))/(2*b)))/(4*Sqrt[a]*Sqrt[a - b]))
3.3.57.3.1 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_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[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]
Int[(x_)^(m_.)*((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[1/2^n Int[x^m*(2*a - b + b*Cosh[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]))
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. \(918\) vs. \(2(351)=702\).
Time = 1.74 (sec) , antiderivative size = 919, normalized size of antiderivative = 2.09
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 ^3+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^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^3-1/2/(-2*((a-b)*a)^(1/2)-2*a +b)*x^4-1/2/((a-b)*a)^(1/2)/(-2*((a-b)*a)^(1/2)-2*a+b)*x^4*a+1/4/((a-b)*a) ^(1/2)/(-2*((a-b)*a)^(1/2)-2*a+b)*b*x^4+3/2/(-2*((a-b)*a)^(1/2)-2*a+b)*pol ylog(2,b*exp(2*x)/(-2*((a-b)*a)^(1/2)-2*a+b))*x^2+3/2/((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))*a* x^2-3/4/((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^2-3/2/(-2*((a-b)*a)^(1/2)-2*a+b)*polylog(3,b *exp(2*x)/(-2*((a-b)*a)^(1/2)-2*a+b))*x-3/2/((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*x+3/4/((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*x+3/4/(-2*((a-b)*a)^(1/2)-2*a+b)*polylog(4,b*exp(2*x)/(-2* ((a-b)*a)^(1/2)-2*a+b))+3/4/((a-b)*a)^(1/2)/(-2*((a-b)*a)^(1/2)-2*a+b)*pol ylog(4,b*exp(2*x)/(-2*((a-b)*a)^(1/2)-2*a+b))*a-3/8/((a-b)*a)^(1/2)/(-2*(( a-b)*a)^(1/2)-2*a+b)*polylog(4,b*exp(2*x)/(-2*((a-b)*a)^(1/2)-2*a+b))*b+1/ 2/((a-b)*a)^(1/2)*x^3*ln(1-b*exp(2*x)/(2*((a-b)*a)^(1/2)-2*a+b))-1/4/((a-b )*a)^(1/2)*x^4+3/4/((a-b)*a)^(1/2)*x^2*polylog(2,b*exp(2*x)/(2*((a-b)*a)^( 1/2)-2*a+b))-3/4/((a-b)*a)^(1/2)*x*polylog(3,b*exp(2*x)/(2*((a-b)*a)^(1...
Leaf count of result is larger than twice the leaf count of optimal. 1655 vs. \(2 (339) = 678\).
Time = 0.30 (sec) , antiderivative size = 1655, normalized size of antiderivative = 3.77 \[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\text {Too large to display} \]
-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)*c osh(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))*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) -...
\[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\int \frac {x^{3}}{a + b \sinh ^{2}{\left (x \right )}}\, dx \]
\[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{3}}{b \sinh \left (x\right )^{2} + a} \,d x } \]
\[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{3}}{b \sinh \left (x\right )^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x^3}{a+b \sinh ^2(x)} \, dx=\int \frac {x^3}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \]