3.3.33 \(\int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [233]

Optimal. Leaf size=712 \[ -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2} \]

[Out]

-3/4*e*f^2*x/b/d^2-3/8*f^3*x^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f-1/8*(f*x+e)^4/b/f-6*a*f^2*(f*x+e)*cosh(d*x+c)/b^2
/d^3-a*(f*x+e)^3*cosh(d*x+c)/b^2/d+6*a*f^3*sinh(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*sinh(d*x+c)/b^2/d^2+3/4*f^2*(f*
x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d^3+1/2*(f*x+e)^3*cosh(d*x+c)*sinh(d*x+c)/b/d-3/8*f^3*sinh(d*x+c)^2/b/d^4-3/4*f
*(f*x+e)^2*sinh(d*x+c)^2/b/d^2-a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d/(a^2+b^2)^(1/2)+a^3*
(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d/(a^2+b^2)^(1/2)-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+
c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2/(a^2+b^2)^(1/2)+3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))
/b^3/d^2/(a^2+b^2)^(1/2)+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^3/(a^2+b^2)^(1/2
)-6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^3/(a^2+b^2)^(1/2)-6*a^3*f^3*polylog(4,-
b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^4/(a^2+b^2)^(1/2)+6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)
))/b^3/d^4/(a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.84, antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5676, 3392, 32, 3391, 3377, 2717, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4 \sqrt {a^2+b^2}}+\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4 \sqrt {a^2+b^2}}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2221

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]
- 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 2296

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 2320

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 2611

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 2717

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

Rule 3377

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 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3403

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 5676

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 6724

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 6744

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 {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {a \int (e+f x)^3 \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x)^3 \, dx}{2 b}+\frac {\left (3 f^2\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {a^2 \int (e+f x)^3 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(3 a f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2 d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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^3}-\frac {\left (6 a f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\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^2 \sqrt {a^2+b^2}}+\frac {\left (6 a f^3\right ) \int \cosh (c+d x) \, dx}{b^2 d^3}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (3 a^3 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^3 \sqrt {a^2+b^2} d}-\frac {\left (3 a^3 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^3 \sqrt {a^2+b^2} d}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (6 a^3 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^3 \sqrt {a^2+b^2} d^2}-\frac {\left (6 a^3 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^3 \sqrt {a^2+b^2} d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 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^3 \sqrt {a^2+b^2} d^3}+\frac {\left (6 a^3 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^3 \sqrt {a^2+b^2} d^3}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 f^3\right ) \text {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^3 \sqrt {a^2+b^2} d^4}+\frac {\left (6 a^3 f^3\right ) \text {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^3 \sqrt {a^2+b^2} d^4}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1948\) vs. \(2(712)=1424\).
time = 7.79, size = 1948, normalized size = 2.74 \begin {gather*} -\frac {\left (-2 a^2+b^2\right ) e^3 x}{2 b^3}-\frac {3 \left (-2 a^2+b^2\right ) e^2 f x^2}{4 b^3}-\frac {\left (-2 a^2+b^2\right ) e f^2 x^3}{2 b^3}-\frac {\left (-2 a^2+b^2\right ) f^3 x^4}{8 b^3}+\frac {a^3 \left (2 d^3 e^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-3 \sqrt {a^2+b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {a^2+b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\sqrt {a^2+b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {a^2+b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {a^2+b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\sqrt {a^2+b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {a^2+b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {a^2+b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {a^2+b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {a^2+b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {a^2+b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {a^2+b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {a^2+b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {a^2+b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{b^3 \sqrt {a^2+b^2} d^4 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\left (-\frac {a f^3 x^3 \cosh (c)}{2 b^2 d}+\frac {a f^3 x^3 \sinh (c)}{2 b^2 d}+\left (d^3 e^3+3 d^2 e^2 f+6 d e f^2+6 f^3\right ) \left (-\frac {a \cosh (c)}{2 b^2 d^4}+\frac {a \sinh (c)}{2 b^2 d^4}\right )+\left (a d^2 e^2 f+2 a d e f^2+2 a f^3\right ) \left (-\frac {3 x \cosh (c)}{2 b^2 d^3}+\frac {3 x \sinh (c)}{2 b^2 d^3}\right )+\left (a d e f^2+a f^3\right ) \left (-\frac {3 x^2 \cosh (c)}{2 b^2 d^2}+\frac {3 x^2 \sinh (c)}{2 b^2 d^2}\right )\right ) (\cosh (d x)-\sinh (d x))+\left (-\frac {a f^3 x^3 \cosh (c)}{2 b^2 d}-\frac {a f^3 x^3 \sinh (c)}{2 b^2 d}+\left (d^3 e^3-3 d^2 e^2 f+6 d e f^2-6 f^3\right ) \left (-\frac {a \cosh (c)}{2 b^2 d^4}-\frac {a \sinh (c)}{2 b^2 d^4}\right )-\frac {3 x^2 \left (a d e f^2 \cosh (c)-a f^3 \cosh (c)+a d e f^2 \sinh (c)-a f^3 \sinh (c)\right )}{2 b^2 d^2}-\frac {3 x \left (a d^2 e^2 f \cosh (c)-2 a d e f^2 \cosh (c)+2 a f^3 \cosh (c)+a d^2 e^2 f \sinh (c)-2 a d e f^2 \sinh (c)+2 a f^3 \sinh (c)\right )}{2 b^2 d^3}\right ) (\cosh (d x)+\sinh (d x))+\left (-\frac {f^3 x^3 \cosh (2 c)}{8 b d}+\frac {f^3 x^3 \sinh (2 c)}{8 b d}+\left (4 d^3 e^3+6 d^2 e^2 f+6 d e f^2+3 f^3\right ) \left (-\frac {\cosh (2 c)}{32 b d^4}+\frac {\sinh (2 c)}{32 b d^4}\right )+\left (2 d^2 e^2 f+2 d e f^2+f^3\right ) \left (-\frac {3 x \cosh (2 c)}{16 b d^3}+\frac {3 x \sinh (2 c)}{16 b d^3}\right )+\left (2 d e f^2+f^3\right ) \left (-\frac {3 x^2 \cosh (2 c)}{16 b d^2}+\frac {3 x^2 \sinh (2 c)}{16 b d^2}\right )\right ) (\cosh (2 d x)-\sinh (2 d x))+\left (\frac {f^3 x^3 \cosh (2 c)}{8 b d}+\frac {f^3 x^3 \sinh (2 c)}{8 b d}+\left (4 d^3 e^3-6 d^2 e^2 f+6 d e f^2-3 f^3\right ) \left (\frac {\cosh (2 c)}{32 b d^4}+\frac {\sinh (2 c)}{32 b d^4}\right )+\frac {3 x^2 \left (2 d e f^2 \cosh (2 c)-f^3 \cosh (2 c)+2 d e f^2 \sinh (2 c)-f^3 \sinh (2 c)\right )}{16 b d^2}+\frac {3 x \left (2 d^2 e^2 f \cosh (2 c)-2 d e f^2 \cosh (2 c)+f^3 \cosh (2 c)+2 d^2 e^2 f \sinh (2 c)-2 d e f^2 \sinh (2 c)+f^3 \sinh (2 c)\right )}{16 b d^3}\right ) (\cosh (2 d x)+\sinh (2 d x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-2*a^2 + b^2)*e^3*x)/b^3 - (3*(-2*a^2 + b^2)*e^2*f*x^2)/(4*b^3) - ((-2*a^2 + b^2)*e*f^2*x^3)/(2*b^3) -
((-2*a^2 + b^2)*f^3*x^4)/(8*b^3) + (a^3*(2*d^3*e^3*Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTanh[(a + b*E^(c + d*x))/Sqrt[
a^2 + b^2]] - 3*Sqrt[a^2 + b^2]*d^3*e^2*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])]
- 3*Sqrt[a^2 + b^2]*d^3*e*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - Sqrt[a
^2 + b^2]*d^3*E^c*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 3*Sqrt[a^2 + b^2]*d
^3*e^2*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 3*Sqrt[a^2 + b^2]*d^3*e*E^c*f^
2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + Sqrt[a^2 + b^2]*d^3*E^c*f^3*x^3*Log[1 +
(b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*Sqrt[a^2 + b^2]*d^2*E^c*f*(e + f*x)^2*PolyLog[2, -
((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 3*Sqrt[a^2 + b^2]*d^2*E^c*f*(e + f*x)^2*PolyLog[2,
-((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[a^2 + b^2]*d*e*E^c*f^2*PolyLog[3, -((b*E^(2
*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[a^2 + b^2]*d*E^c*f^3*x*PolyLog[3, -((b*E^(2*c + d*x)
)/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[a^2 + b^2]*d*e*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c
+ Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[a^2 + b^2]*d*E^c*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a
^2 + b^2)*E^(2*c)]))] - 6*Sqrt[a^2 + b^2]*E^c*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(
2*c)]))] + 6*Sqrt[a^2 + b^2]*E^c*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(b
^3*Sqrt[a^2 + b^2]*d^4*Sqrt[(a^2 + b^2)*E^(2*c)]) + (-1/2*(a*f^3*x^3*Cosh[c])/(b^2*d) + (a*f^3*x^3*Sinh[c])/(2
*b^2*d) + (d^3*e^3 + 3*d^2*e^2*f + 6*d*e*f^2 + 6*f^3)*(-1/2*(a*Cosh[c])/(b^2*d^4) + (a*Sinh[c])/(2*b^2*d^4)) +
(a*d^2*e^2*f + 2*a*d*e*f^2 + 2*a*f^3)*((-3*x*Cosh[c])/(2*b^2*d^3) + (3*x*Sinh[c])/(2*b^2*d^3)) + (a*d*e*f^2 +
a*f^3)*((-3*x^2*Cosh[c])/(2*b^2*d^2) + (3*x^2*Sinh[c])/(2*b^2*d^2)))*(Cosh[d*x] - Sinh[d*x]) + (-1/2*(a*f^3*x
^3*Cosh[c])/(b^2*d) - (a*f^3*x^3*Sinh[c])/(2*b^2*d) + (d^3*e^3 - 3*d^2*e^2*f + 6*d*e*f^2 - 6*f^3)*(-1/2*(a*Cos
h[c])/(b^2*d^4) - (a*Sinh[c])/(2*b^2*d^4)) - (3*x^2*(a*d*e*f^2*Cosh[c] - a*f^3*Cosh[c] + a*d*e*f^2*Sinh[c] - a
*f^3*Sinh[c]))/(2*b^2*d^2) - (3*x*(a*d^2*e^2*f*Cosh[c] - 2*a*d*e*f^2*Cosh[c] + 2*a*f^3*Cosh[c] + a*d^2*e^2*f*S
inh[c] - 2*a*d*e*f^2*Sinh[c] + 2*a*f^3*Sinh[c]))/(2*b^2*d^3))*(Cosh[d*x] + Sinh[d*x]) + (-1/8*(f^3*x^3*Cosh[2*
c])/(b*d) + (f^3*x^3*Sinh[2*c])/(8*b*d) + (4*d^3*e^3 + 6*d^2*e^2*f + 6*d*e*f^2 + 3*f^3)*(-1/32*Cosh[2*c]/(b*d^
4) + Sinh[2*c]/(32*b*d^4)) + (2*d^2*e^2*f + 2*d*e*f^2 + f^3)*((-3*x*Cosh[2*c])/(16*b*d^3) + (3*x*Sinh[2*c])/(1
6*b*d^3)) + (2*d*e*f^2 + f^3)*((-3*x^2*Cosh[2*c])/(16*b*d^2) + (3*x^2*Sinh[2*c])/(16*b*d^2)))*(Cosh[2*d*x] - S
inh[2*d*x]) + ((f^3*x^3*Cosh[2*c])/(8*b*d) + (f^3*x^3*Sinh[2*c])/(8*b*d) + (4*d^3*e^3 - 6*d^2*e^2*f + 6*d*e*f^
2 - 3*f^3)*(Cosh[2*c]/(32*b*d^4) + Sinh[2*c]/(32*b*d^4)) + (3*x^2*(2*d*e*f^2*Cosh[2*c] - f^3*Cosh[2*c] + 2*d*e
*f^2*Sinh[2*c] - f^3*Sinh[2*c]))/(16*b*d^2) + (3*x*(2*d^2*e^2*f*Cosh[2*c] - 2*d*e*f^2*Cosh[2*c] + f^3*Cosh[2*c
] + 2*d^2*e^2*f*Sinh[2*c] - 2*d*e*f^2*Sinh[2*c] + f^3*Sinh[2*c]))/(16*b*d^3))*(Cosh[2*d*x] + Sinh[2*d*x])

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Maple [F]
time = 0.99, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2
)*b^3*d) + (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 4*(2*a^2 - b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x -
c) + b*e^(-2*d*x - 2*c))/(b^2*d))*e^3 + 1/32*(4*(2*a^2*d^4*f^3*e^(2*c) - b^2*d^4*f^3*e^(2*c))*x^4 + 16*(2*a^2
*d^4*f^2*e^(2*c) - b^2*d^4*f^2*e^(2*c))*x^3*e + 24*(2*a^2*d^4*f*e^(2*c) - b^2*d^4*f*e^(2*c))*x^2*e^2 + (4*b^2*
d^3*f^3*x^3*e^(4*c) - 3*b^2*f^3*e^(4*c) - 6*b^2*d^2*f*e^(4*c + 2) + 6*b^2*d*f^2*e^(4*c + 1) - 6*(b^2*d^2*f^3*e
^(4*c) - 2*b^2*d^3*f^2*e^(4*c + 1))*x^2 + 6*(b^2*d*f^3*e^(4*c) + 2*b^2*d^3*f*e^(4*c + 2) - 2*b^2*d^2*f^2*e^(4*
c + 1))*x)*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) - 6*a*b*f^3*e^(3*c) - 3*a*b*d^2*f*e^(3*c + 2) + 6*a*b*d*f^2
*e^(3*c + 1) - 3*(a*b*d^2*f^3*e^(3*c) - a*b*d^3*f^2*e^(3*c + 1))*x^2 + 3*(2*a*b*d*f^3*e^(3*c) + a*b*d^3*f*e^(3
*c + 2) - 2*a*b*d^2*f^2*e^(3*c + 1))*x)*e^(d*x) - 16*(a*b*d^3*f^3*x^3*e^c + 3*a*b*d^2*f*e^(c + 2) + 6*a*b*d*f^
2*e^(c + 1) + 6*a*b*f^3*e^c + 3*(a*b*d^3*f^2*e^(c + 1) + a*b*d^2*f^3*e^c)*x^2 + 3*(a*b*d^3*f*e^(c + 2) + 2*a*b
*d^2*f^2*e^(c + 1) + 2*a*b*d*f^3*e^c)*x)*e^(-d*x) - (4*b^2*d^3*f^3*x^3 + 6*b^2*d^2*f*e^2 + 6*b^2*d*f^2*e + 3*b
^2*f^3 + 6*(2*b^2*d^3*f^2*e + b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*f*e^2 + 2*b^2*d^2*f^2*e + b^2*d*f^3)*x)*e^(-2*d*
x))*e^(-2*c)/(b^3*d^4) - integrate(2*(a^3*f^3*x^3*e^c + 3*a^3*f^2*x^2*e^(c + 1) + 3*a^3*f*x*e^(c + 2))*e^(d*x)
/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9525 vs. \(2 (669) = 1338\).
time = 0.45, size = 9525, normalized size = 13.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(4*(a^2*b^2 + b^4)*d^3*f^3*x^3 + 6*(a^2*b^2 + b^4)*d^2*f^3*x^2 + 4*(a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(a^
2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(a^2*b^2 + b^4)*d*f^3*x - (4*(a^2*b^2 + b^4)*d^3*f^3*x^3 - 6*(a^2*b^2 + b^4)*d^
2*f^3*x^2 + 4*(a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(a^2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(a^2*b^2 + b^4)*d*f^3*x - 3*
(a^2*b^2 + b^4)*f^3 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x - (a^2*b^2 + b^4)*d^2*f)*cosh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d
^3*f*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1) - (a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 -
2*(a^2*b^2 + b^4)*d^2*f^2*x + (a^2*b^2 + b^4)*d*f^2)*cosh(1) + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(a^2*b^2
+ b^4)*d^2*f^2*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1)^2 + (a^2*b^2 + b^4)*d*f^2 + 2*(2*(a^2*b^2 + b^4)*d^3*f*x - (a
^2*b^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^4 - (4*(a^2*b^2 + b^4)*d^3*f^3*x^3 - 6*(a^2*b^2 + b^4)*d^
2*f^3*x^2 + 4*(a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(a^2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(a^2*b^2 + b^4)*d*f^3*x - 3*
(a^2*b^2 + b^4)*f^3 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x - (a^2*b^2 + b^4)*d^2*f)*cosh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d
^3*f*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1) - (a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 -
2*(a^2*b^2 + b^4)*d^2*f^2*x + (a^2*b^2 + b^4)*d*f^2)*cosh(1) + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(a^2*b^2
+ b^4)*d^2*f^2*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1)^2 + (a^2*b^2 + b^4)*d*f^2 + 2*(2*(a^2*b^2 + b^4)*d^3*f*x - (a
^2*b^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^4 + 3*(a^2*b^2 + b^4)*f^3 + 16*((a^3*b + a*b^3)*d^3*f^3*x
^3 - 3*(a^3*b + a*b^3)*d^2*f^3*x^2 + (a^3*b + a*b^3)*d^3*cosh(1)^3 + (a^3*b + a*b^3)*d^3*sinh(1)^3 + 6*(a^3*b
+ a*b^3)*d*f^3*x - 6*(a^3*b + a*b^3)*f^3 + 3*((a^3*b + a*b^3)*d^3*f*x - (a^3*b + a*b^3)*d^2*f)*cosh(1)^2 + 3*(
(a^3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*d^3*cosh(1) - (a^3*b + a*b^3)*d^2*f)*sinh(1)^2 + 3*((a^3*b + a*b^3)*
d^3*f^2*x^2 - 2*(a^3*b + a*b^3)*d^2*f^2*x + 2*(a^3*b + a*b^3)*d*f^2)*cosh(1) + 3*((a^3*b + a*b^3)*d^3*f^2*x^2
- 2*(a^3*b + a*b^3)*d^2*f^2*x + (a^3*b + a*b^3)*d^3*cosh(1)^2 + 2*(a^3*b + a*b^3)*d*f^2 + 2*((a^3*b + a*b^3)*d
^3*f*x - (a^3*b + a*b^3)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^3 + 4*(4*(a^3*b + a*b^3)*d^3*f^3*x^3 - 12*(a^3
*b + a*b^3)*d^2*f^3*x^2 + 4*(a^3*b + a*b^3)*d^3*cosh(1)^3 + 4*(a^3*b + a*b^3)*d^3*sinh(1)^3 + 24*(a^3*b + a*b^
3)*d*f^3*x - 24*(a^3*b + a*b^3)*f^3 + 12*((a^3*b + a*b^3)*d^3*f*x - (a^3*b + a*b^3)*d^2*f)*cosh(1)^2 + 12*((a^
3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*d^3*cosh(1) - (a^3*b + a*b^3)*d^2*f)*sinh(1)^2 + 12*((a^3*b + a*b^3)*d^
3*f^2*x^2 - 2*(a^3*b + a*b^3)*d^2*f^2*x + 2*(a^3*b + a*b^3)*d*f^2)*cosh(1) - (4*(a^2*b^2 + b^4)*d^3*f^3*x^3 -
6*(a^2*b^2 + b^4)*d^2*f^3*x^2 + 4*(a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(a^2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(a^2*b^2
+ b^4)*d*f^3*x - 3*(a^2*b^2 + b^4)*f^3 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x - (a^2*b^2 + b^4)*d^2*f)*cosh(1)^2 + 6*
(2*(a^2*b^2 + b^4)*d^3*f*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1) - (a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 + 6*(2*(a^2*b^2
+ b^4)*d^3*f^2*x^2 - 2*(a^2*b^2 + b^4)*d^2*f^2*x + (a^2*b^2 + b^4)*d*f^2)*cosh(1) + 6*(2*(a^2*b^2 + b^4)*d^3*f
^2*x^2 - 2*(a^2*b^2 + b^4)*d^2*f^2*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1)^2 + (a^2*b^2 + b^4)*d*f^2 + 2*(2*(a^2*b^2
+ b^4)*d^3*f*x - (a^2*b^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 12*((a^3*b + a*b^3)*d^3*f^2*x^2 - 2
*(a^3*b + a*b^3)*d^2*f^2*x + (a^3*b + a*b^3)*d^3*cosh(1)^2 + 2*(a^3*b + a*b^3)*d*f^2 + 2*((a^3*b + a*b^3)*d^3*
f*x - (a^3*b + a*b^3)*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^3 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x + (a^2*b^2 + b^4
)*d^2*f)*cosh(1)^2 - 4*((2*a^4 + a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(2*a^4 + a^2*b^2 - b^4)*d^4*f^2*x^3*cosh(1) +
6*(2*a^4 + a^2*b^2 - b^4)*d^4*f*x^2*cosh(1)^2 + 4*(2*a^4 + a^2*b^2 - b^4)*d^4*x*cosh(1)^3 + 4*(2*a^4 + a^2*b^2
- b^4)*d^4*x*sinh(1)^3 + 6*((2*a^4 + a^2*b^2 - b^4)*d^4*f*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^4*x*cosh(1))*sinh
(1)^2 + 4*((2*a^4 + a^2*b^2 - b^4)*d^4*f^2*x^3 + 3*(2*a^4 + a^2*b^2 - b^4)*d^4*f*x^2*cosh(1) + 3*(2*a^4 + a^2*
b^2 - b^4)*d^4*x*cosh(1)^2)*sinh(1))*cosh(d*x + c)^2 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x + 2*(a^2*b^2 + b^4)*d^3*co
sh(1) + (a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 - 2*(2*(2*a^4 + a^2*b^2 - b^4)*d^4*f^3*x^4 + 8*(2*a^4 + a^2*b^2 - b^4
)*d^4*f^2*x^3*cosh(1) + 12*(2*a^4 + a^2*b^2 - b^4)*d^4*f*x^2*cosh(1)^2 + 8*(2*a^4 + a^2*b^2 - b^4)*d^4*x*cosh(
1)^3 + 8*(2*a^4 + a^2*b^2 - b^4)*d^4*x*sinh(1)^3 + 3*(4*(a^2*b^2 + b^4)*d^3*f^3*x^3 - 6*(a^2*b^2 + b^4)*d^2*f^
3*x^2 + 4*(a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(a^2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(a^2*b^2 + b^4)*d*f^3*x - 3*(a^2
*b^2 + b^4)*f^3 + 6*(2*(a^2*b^2 + b^4)*d^3*f*x - (a^2*b^2 + b^4)*d^2*f)*cosh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d^3*f
*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1) - (a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(
a^2*b^2 + b^4)*d^2*f^2*x + (a^2*b^2 + b^4)*d*f^2)*cosh(1) + 6*(2*(a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(a^2*b^2 + b^
4)*d^2*f^2*x + 2*(a^2*b^2 + b^4)*d^3*cosh(1)^2 + (a^2*b^2 + b^4)*d*f^2 + 2*(2*(a^2*b^2 + b^4)*d^3*f*x - (a^2*b
^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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