3.228 \(\int \frac{x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\)

Optimal. Leaf size=495 \[ \frac{3 x^2 \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 x^2 \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}-\frac{6 x \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 x \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^4}-\frac{6 \sqrt{a^2-b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^4}+\frac{x^3 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x^3 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^4}{4 b^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{6 x \sinh (c+d x)}{b d^3}-\frac{6 \cosh (c+d x)}{b d^4}+\frac{x^3 \sinh (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.838589, antiderivative size = 495, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {5566, 30, 3296, 2638, 3320, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 x^2 \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}-\frac{6 x \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 x \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^4}-\frac{6 \sqrt{a^2-b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^4}+\frac{x^3 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x^3 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^4}{4 b^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{6 x \sinh (c+d x)}{b d^3}-\frac{6 \cosh (c+d x)}{b d^4}+\frac{x^3 \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

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

Rule 5566

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_))/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(
n - 2)*Cosh[c + d*x], x], x] + Dist[(a^2 - b^2)/b^2, Int[((e + f*x)^m*Sinh[c + d*x]^(n - 2))/(a + b*Cosh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3320

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

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 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]

Rubi steps

\begin{align*} \int \frac{x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac{a \int x^3 \, dx}{b^2}+\frac{\int x^3 \cosh (c+d x) \, dx}{b}+\frac{\left (a^2-b^2\right ) \int \frac{x^3}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac{a x^4}{4 b^2}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{\left (2 \left (a^2-b^2\right )\right ) \int \frac{e^{c+d x} x^3}{b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac{3 \int x^2 \sinh (c+d x) \, dx}{b d}\\ &=-\frac{a x^4}{4 b^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{\left (2 \sqrt{a^2-b^2}\right ) \int \frac{e^{c+d x} x^3}{2 a-2 \sqrt{a^2-b^2}+2 b e^{c+d x}} \, dx}{b}-\frac{\left (2 \sqrt{a^2-b^2}\right ) \int \frac{e^{c+d x} x^3}{2 a+2 \sqrt{a^2-b^2}+2 b e^{c+d x}} \, dx}{b}+\frac{6 \int x \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac{a x^4}{4 b^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{x^3 \sinh (c+d x)}{b d}-\frac{6 \int \sinh (c+d x) \, dx}{b d^3}-\frac{\left (3 \sqrt{a^2-b^2}\right ) \int x^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d}+\frac{\left (3 \sqrt{a^2-b^2}\right ) \int x^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d}\\ &=-\frac{a x^4}{4 b^2}-\frac{6 \cosh (c+d x)}{b d^4}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{x^3 \sinh (c+d x)}{b d}-\frac{\left (6 \sqrt{a^2-b^2}\right ) \int x \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d^2}+\frac{\left (6 \sqrt{a^2-b^2}\right ) \int x \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d^2}\\ &=-\frac{a x^4}{4 b^2}-\frac{6 \cosh (c+d x)}{b d^4}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{\left (6 \sqrt{a^2-b^2}\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d^3}-\frac{\left (6 \sqrt{a^2-b^2}\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b^2 d^3}\\ &=-\frac{a x^4}{4 b^2}-\frac{6 \cosh (c+d x)}{b d^4}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{\left (6 \sqrt{a^2-b^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{b x}{-a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}-\frac{\left (6 \sqrt{a^2-b^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}\\ &=-\frac{a x^4}{4 b^2}-\frac{6 \cosh (c+d x)}{b d^4}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2-b^2} x^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2-b^2} x^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} x \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2-b^2} \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^4}-\frac{6 \sqrt{a^2-b^2} \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^4}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{x^3 \sinh (c+d x)}{b d}\\ \end{align*}

Mathematica [A]  time = 1.39152, size = 386, normalized size = 0.78 \[ \frac{4 \sqrt{a^2-b^2} \left (3 d^2 x^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )-3 d^2 x^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )-6 d x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )+6 d x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )+6 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )-6 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )+d^3 x^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )-d^3 x^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )\right )-a d^4 x^4+4 b \cosh (d x) \left (d x \sinh (c) \left (d^2 x^2+6\right )-3 \cosh (c) \left (d^2 x^2+2\right )\right )+4 b \sinh (d x) \left (d x \cosh (c) \left (d^2 x^2+6\right )-3 \sinh (c) \left (d^2 x^2+2\right )\right )}{4 b^2 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

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

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{a+b\cosh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

[Out]

int(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.21682, size = 2889, normalized size = 5.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a*d^4*x^4*cosh(d*x + c) + 2*b*d^3*x^3 + 6*b*d^2*x^2 + 12*b*d*x - 2*(b*d^3*x^3 - 3*b*d^2*x^2 + 6*b*d*x -
6*b)*cosh(d*x + c)^2 - 2*(b*d^3*x^3 - 3*b*d^2*x^2 + 6*b*d*x - 6*b)*sinh(d*x + c)^2 - 12*(b*d^2*x^2*cosh(d*x +
c) + b*d^2*x^2*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + 12*(b*d^2*x^2*cosh(d*x + c) + b*d^2*x^2*sinh(d*x + c
))*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(
(a^2 - b^2)/b^2) + b)/b + 1) - 4*(b*c^3*cosh(d*x + c) + b*c^3*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log(2*b*cos
h(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) + 4*(b*c^3*cosh(d*x + c) + b*c^3*sinh(d*x +
c))*sqrt((a^2 - b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) - 4*((b
*d^3*x^3 + b*c^3)*cosh(d*x + c) + (b*d^3*x^3 + b*c^3)*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log((a*cosh(d*x + c
) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) + 4*((b*d^3*x^3 + b*c^
3)*cosh(d*x + c) + (b*d^3*x^3 + b*c^3)*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) - 24*(b*cosh(d*x + c) + b*sinh(d*x +
c))*sqrt((a^2 - b^2)/b^2)*polylog(4, -(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))
*sqrt((a^2 - b^2)/b^2))/b) + 24*(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*polylog(4, -(a*cosh(
d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2))/b) + 24*(b*d*x*cosh(d*
x + c) + b*d*x*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*polylog(3, -(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d
*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2))/b) - 24*(b*d*x*cosh(d*x + c) + b*d*x*sinh(d*x + c))*sqrt((a^
2 - b^2)/b^2)*polylog(3, -(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 -
b^2)/b^2))/b) + (a*d^4*x^4 - 4*(b*d^3*x^3 - 3*b*d^2*x^2 + 6*b*d*x - 6*b)*cosh(d*x + c))*sinh(d*x + c) + 12*b)
/(b^2*d^4*cosh(d*x + c) + b^2*d^4*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)