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

Optimal. Leaf size=551 \[ -\frac{6 a^2 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3 \sqrt{a^2+b^2}}+\frac{6 a^2 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3 \sqrt{a^2+b^2}}+\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2 \sqrt{a^2+b^2}}-\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2 \sqrt{a^2+b^2}}+\frac{6 a^2 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^4 \sqrt{a^2+b^2}}-\frac{6 a^2 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^4 \sqrt{a^2+b^2}}+\frac{a^2 (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d \sqrt{a^2+b^2}}-\frac{a^2 (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d \sqrt{a^2+b^2}}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{6 f^3 \sinh (c+d x)}{b d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.02733, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5557, 3296, 2637, 32, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{6 a^2 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3 \sqrt{a^2+b^2}}+\frac{6 a^2 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3 \sqrt{a^2+b^2}}+\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2 \sqrt{a^2+b^2}}-\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2 \sqrt{a^2+b^2}}+\frac{6 a^2 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^4 \sqrt{a^2+b^2}}-\frac{6 a^2 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^4 \sqrt{a^2+b^2}}+\frac{a^2 (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d \sqrt{a^2+b^2}}-\frac{a^2 (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d \sqrt{a^2+b^2}}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{6 f^3 \sinh (c+d x)}{b d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5557

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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

Mathematica [A]  time = 2.96971, size = 979, normalized size = 1.78 \[ \frac{-a \sqrt{a^2+b^2} f^3 x^4 d^4-4 a \sqrt{a^2+b^2} e f^2 x^3 d^4-6 a \sqrt{a^2+b^2} e^2 f x^2 d^4-4 a \sqrt{a^2+b^2} e^3 x d^4-8 a^2 e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^3+4 b \sqrt{a^2+b^2} e^3 \cosh (c+d x) d^3+4 b \sqrt{a^2+b^2} f^3 x^3 \cosh (c+d x) d^3+12 b \sqrt{a^2+b^2} e f^2 x^2 \cosh (c+d x) d^3+12 b \sqrt{a^2+b^2} e^2 f x \cosh (c+d x) d^3+4 a^2 f^3 x^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+12 a^2 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+12 a^2 e^2 f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-4 a^2 f^3 x^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-12 a^2 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-12 a^2 e^2 f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+12 a^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d^2-12 a^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d^2-12 b \sqrt{a^2+b^2} f^3 x^2 \sinh (c+d x) d^2-12 b \sqrt{a^2+b^2} e^2 f \sinh (c+d x) d^2-24 b \sqrt{a^2+b^2} e f^2 x \sinh (c+d x) d^2+24 b \sqrt{a^2+b^2} e f^2 \cosh (c+d x) d+24 b \sqrt{a^2+b^2} f^3 x \cosh (c+d x) d-24 a^2 e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d-24 a^2 f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+24 a^2 e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+24 a^2 f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+24 a^2 f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-24 a^2 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )-24 b \sqrt{a^2+b^2} f^3 \sinh (c+d x)}{4 b^2 \sqrt{a^2+b^2} d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*Sqrt[a^2 + b^2]*d^4*e^3*x - 6*a*Sqrt[a^2 + b^2]*d^4*e^2*f*x^2 - 4*a*Sqrt[a^2 + b^2]*d^4*e*f^2*x^3 - a*Sq
rt[a^2 + b^2]*d^4*f^3*x^4 - 8*a^2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 4*b*Sqrt[a^2 + b^2]*d
^3*e^3*Cosh[c + d*x] + 24*b*Sqrt[a^2 + b^2]*d*e*f^2*Cosh[c + d*x] + 12*b*Sqrt[a^2 + b^2]*d^3*e^2*f*x*Cosh[c +
d*x] + 24*b*Sqrt[a^2 + b^2]*d*f^3*x*Cosh[c + d*x] + 12*b*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*Cosh[c + d*x] + 4*b*Sqr
t[a^2 + b^2]*d^3*f^3*x^3*Cosh[c + d*x] + 12*a^2*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 1
2*a^2*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 4*a^2*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))
/(a - Sqrt[a^2 + b^2])] - 12*a^2*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 12*a^2*d^3*e*f^2
*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 4*a^2*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2])] + 12*a^2*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 12*a^2*d^2*f*(e + f*x
)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 24*a^2*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqr
t[a^2 + b^2])] - 24*a^2*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 24*a^2*d*e*f^2*PolyLog[3,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 24*a^2*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)] + 24*a^2*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 24*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(
a + Sqrt[a^2 + b^2]))] - 12*b*Sqrt[a^2 + b^2]*d^2*e^2*f*Sinh[c + d*x] - 24*b*Sqrt[a^2 + b^2]*f^3*Sinh[c + d*x]
- 24*b*Sqrt[a^2 + b^2]*d^2*e*f^2*x*Sinh[c + d*x] - 12*b*Sqrt[a^2 + b^2]*d^2*f^3*x^2*Sinh[c + d*x])/(4*b^2*Sqr
t[a^2 + b^2]*d^4)

________________________________________________________________________________________

Maple [F]  time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 3.19243, size = 5847, normalized size = 10.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(a^2*b + b^3)*d^3*f^3*x^3 + 2*(a^2*b + b^3)*d^3*e^3 + 6*(a^2*b + b^3)*d^2*e^2*f + 12*(a^2*b + b^3)*d*e*
f^2 + 12*(a^2*b + b^3)*f^3 + 6*((a^2*b + b^3)*d^3*e*f^2 + (a^2*b + b^3)*d^2*f^3)*x^2 + 2*((a^2*b + b^3)*d^3*f^
3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3*
((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2
+ 2*(a^2*b + b^3)*d*f^3)*x)*cosh(d*x + c)^2 + 2*((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b
+ b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)
*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x)*sinh(d*x +
c)^2 + 12*((a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*
a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*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*((a^2*b*d^2*f^3*x^2 + 2*a^2*
b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*s
inh(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*((a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b
*c^3*f^3)*cosh(d*x + c) + (a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^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*
((a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*cosh(d*x + c) + (a^2*b*d^3*e^3 -
3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^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*((a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2
+ 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*cosh(d*x + c) + (a^2*b*d^3
*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3
*f^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*((a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x
+ 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*cosh(d*x + c) + (a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3
*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*sinh(d*x + c))*s
qrt((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*(a^2*b*f^3*cosh(d*x + c) + a^2*b*f^3*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*(a^2*b*
f^3*cosh(d*x + c) + a^2*b*f^3*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*((a^2*b*d*f^3*x + a^2*b*d*e*f^2)*cosh(
d*x + c) + (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*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*((a^2*b*d*f^3*x + a^2*b*d
*e*f^2)*cosh(d*x + c) + (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cos
h(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) + 6*((a^2*b + b^3
)*d^3*e^2*f + 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x - ((a^3 + a*b^2)*d^4*f^3*x^4 + 4*(a^3 + a*b
^2)*d^4*e*f^2*x^3 + 6*(a^3 + a*b^2)*d^4*e^2*f*x^2 + 4*(a^3 + a*b^2)*d^4*e^3*x)*cosh(d*x + c) - ((a^3 + a*b^2)*
d^4*f^3*x^4 + 4*(a^3 + a*b^2)*d^4*e*f^2*x^3 + 6*(a^3 + a*b^2)*d^4*e^2*f*x^2 + 4*(a^3 + a*b^2)*d^4*e^3*x - 4*((
a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^
2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2
*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^2*b^2 + b^4)*d^4*cosh(d*x +
c) + (a^2*b^2 + b^4)*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((f*x+e)**3*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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