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

Optimal. Leaf size=586 \[ \frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^4}-\frac {6 x \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 x \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {3 x^2 \left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 x^2 \left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {x^3 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^3 d}+\frac {x^3 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^3 d}-\frac {x^4 \left (a^2-b^2\right )}{4 b^3}+\frac {6 a \sinh (c+d x)}{b^2 d^4}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {a x^3 \cosh (c+d x)}{b^2 d}-\frac {3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}-\frac {3 x^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.69, antiderivative size = 586, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5566, 3296, 2637, 5372, 3311, 30, 2635, 8, 5562, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \left (a^2-b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 x^2 \left (a^2-b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^2}-\frac {6 x \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 x \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^3}+\frac {6 \left (a^2-b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^4}+\frac {x^3 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^3 d}+\frac {x^3 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^3 d}-\frac {x^4 \left (a^2-b^2\right )}{4 b^3}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}+\frac {6 a \sinh (c+d x)}{b^2 d^4}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}-\frac {3 x^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}-\frac {3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(8*b*d^3) + x^3/(4*b*d) - ((a^2 - b^2)*x^4)/(4*b^3) - (6*a*x*Cosh[c + d*x])/(b^2*d^3) - (a*x^3*Cosh[c +
d*x])/(b^2*d) + ((a^2 - b^2)*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^3*d) + ((a^2 - b^2)*x^3*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + (3*(a^2 - b^2)*x^2*PolyLog[2, -((b*E^(c + d*x))/(a - S
qrt[a^2 - b^2]))])/(b^3*d^2) + (3*(a^2 - b^2)*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d
^2) - (6*(a^2 - b^2)*x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^3) - (6*(a^2 - b^2)*x*Poly
Log[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^3) + (6*(a^2 - b^2)*PolyLog[4, -((b*E^(c + d*x))/(a -
Sqrt[a^2 - b^2]))])/(b^3*d^4) + (6*(a^2 - b^2)*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^4
) + (6*a*Sinh[c + d*x])/(b^2*d^4) + (3*a*x^2*Sinh[c + d*x])/(b^2*d^2) - (3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b*d
^4) - (3*x^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (3*x*Sinh[c + d*x]^2)/(4*b*d^3) + (x^3*Sinh[c + d*x]^2)/
(2*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

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 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 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 2635

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

Rule 2637

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

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 3311

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 5372

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m -
n + 1)*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 5562

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

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 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]

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]

Rubi steps

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

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Mathematica [A]  time = 11.57, size = 1082, normalized size = 1.85 \[ -\frac {(b-a) (a+b) \cosh \left (\frac {c}{2}\right ) \text {sech}(c) \sinh \left (\frac {c}{2}\right ) x^4}{2 b^3}-\frac {a \cosh (d x) \left (d^3 \cosh (c) x^3-3 d^2 \sinh (c) x^2+6 d \cosh (c) x-6 \sinh (c)\right )}{b^2 d^4}+\frac {\cosh (2 d x) \left (4 d^3 \cosh (2 c) x^3-6 d^2 \sinh (2 c) x^2+6 d \cosh (2 c) x-3 \sinh (2 c)\right )}{16 b d^4}-\frac {a \left (d^3 \sinh (c) x^3-3 d^2 \cosh (c) x^2+6 d \sinh (c) x-6 \cosh (c)\right ) \sinh (d x)}{b^2 d^4}+\frac {\left (4 d^3 \sinh (2 c) x^3-6 d^2 \cosh (2 c) x^2+6 d \sinh (2 c) x-3 \cosh (2 c)\right ) \sinh (2 d x)}{16 b d^4}+\frac {\left (b^2-a^2\right ) \left (-x^4+\frac {2 b^2 \left (d^3 \log \left (\frac {\left (a-\sqrt {a^2-b^2}\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}+1\right ) x^3-3 d^2 \text {Li}_2\left (\frac {\left (\sqrt {a^2-b^2}-a\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}\right ) x^2-6 d \text {Li}_3\left (\frac {\left (\sqrt {a^2-b^2}-a\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}\right ) x-6 \text {Li}_4\left (\frac {\left (\sqrt {a^2-b^2}-a\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt {a^2-b^2} \left (\sqrt {a^2-b^2}-a\right ) d^4}+\frac {2 b^2 \left (d^3 \log \left (\frac {\left (a+\sqrt {a^2-b^2}\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}+1\right ) x^3-3 d^2 \text {Li}_2\left (\frac {\left (a+\sqrt {a^2-b^2}\right ) (\sinh (c+d x)-\cosh (c+d x))}{b}\right ) x^2-6 d \text {Li}_3\left (\frac {\left (a+\sqrt {a^2-b^2}\right ) (\sinh (c+d x)-\cosh (c+d x))}{b}\right ) x-6 \text {Li}_4\left (\frac {\left (a+\sqrt {a^2-b^2}\right ) (\sinh (c+d x)-\cosh (c+d x))}{b}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt {a^2-b^2} \left (a+\sqrt {a^2-b^2}\right ) d^4}+\frac {2 a \left (d^3 \log \left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a-\sqrt {a^2-b^2}}+1\right ) x^3+3 d^2 \text {Li}_2\left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt {a^2-b^2}-a}\right ) x^2-6 d \text {Li}_3\left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt {a^2-b^2}-a}\right ) x+6 \text {Li}_4\left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt {a^2-b^2}-a}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt {a^2-b^2} d^4}-\frac {2 a \left (d^3 \log \left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2-b^2}}+1\right ) x^3+3 d^2 \text {Li}_2\left (-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2-b^2}}\right ) x^2-6 d \text {Li}_3\left (-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2-b^2}}\right ) x+6 \text {Li}_4\left (-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2-b^2}}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt {a^2-b^2} d^4}\right ) (1-\tanh (c))}{4 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-a + b)*(a + b)*x^4*Cosh[c/2]*Sech[c]*Sinh[c/2])/b^3 - (a*Cosh[d*x]*(6*d*x*Cosh[c] + d^3*x^3*Cosh[c] -
6*Sinh[c] - 3*d^2*x^2*Sinh[c]))/(b^2*d^4) + (Cosh[2*d*x]*(6*d*x*Cosh[2*c] + 4*d^3*x^3*Cosh[2*c] - 3*Sinh[2*c]
- 6*d^2*x^2*Sinh[2*c]))/(16*b*d^4) - (a*(-6*Cosh[c] - 3*d^2*x^2*Cosh[c] + 6*d*x*Sinh[c] + d^3*x^3*Sinh[c])*Sin
h[d*x])/(b^2*d^4) + ((-3*Cosh[2*c] - 6*d^2*x^2*Cosh[2*c] + 6*d*x*Sinh[2*c] + 4*d^3*x^3*Sinh[2*c])*Sinh[2*d*x])
/(16*b*d^4) + ((-a^2 + b^2)*(-x^4 + (2*b^2*(d^3*x^3*Log[1 + ((a - Sqrt[a^2 - b^2])*(Cosh[c + d*x] - Sinh[c + d
*x]))/b] - 3*d^2*x^2*PolyLog[2, ((-a + Sqrt[a^2 - b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 6*d*x*PolyLog[3,
((-a + Sqrt[a^2 - b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 6*PolyLog[4, ((-a + Sqrt[a^2 - b^2])*(Cosh[c +
d*x] - Sinh[c + d*x]))/b])*(1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 - b^2]*(-a + Sqrt[a^2 - b^2])*d^4) + (2*b^2*
(d^3*x^3*Log[1 + ((a + Sqrt[a^2 - b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 3*d^2*x^2*PolyLog[2, ((a + Sqrt[
a^2 - b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b] - 6*d*x*PolyLog[3, ((a + Sqrt[a^2 - b^2])*(-Cosh[c + d*x] + S
inh[c + d*x]))/b] - 6*PolyLog[4, ((a + Sqrt[a^2 - b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b])*(1 + Cosh[2*c] +
Sinh[2*c]))/(Sqrt[a^2 - b^2]*(a + Sqrt[a^2 - b^2])*d^4) + (2*a*(d^3*x^3*Log[1 + (b*(Cosh[c + d*x] + Sinh[c +
d*x]))/(a - Sqrt[a^2 - b^2])] + 3*d^2*x^2*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 - b^2]
)] - 6*d*x*PolyLog[3, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 - b^2])] + 6*PolyLog[4, (b*(Cosh[c +
d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 - b^2])])*(1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 - b^2]*d^4) - (2*a*(d^3
*x^3*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 - b^2])] + 3*d^2*x^2*PolyLog[2, -((b*(Cosh[c +
d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 - b^2]))] - 6*d*x*PolyLog[3, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + S
qrt[a^2 - b^2]))] + 6*PolyLog[4, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 - b^2]))])*(1 + Cosh[2*c]
+ Sinh[2*c]))/(Sqrt[a^2 - b^2]*d^4))*(1 - Tanh[c]))/(4*b^3)

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fricas [C]  time = 0.53, size = 2025, normalized size = 3.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(4*b^2*d^3*x^3 + 6*b^2*d^2*x^2 + (4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2)*cosh(d*x + c)^4 + (4
*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2)*sinh(d*x + c)^4 + 6*b^2*d*x - 16*(a*b*d^3*x^3 - 3*a*b*d^2*x^
2 + 6*a*b*d*x - 6*a*b)*cosh(d*x + c)^3 - 4*(4*a*b*d^3*x^3 - 12*a*b*d^2*x^2 + 24*a*b*d*x - 24*a*b - (4*b^2*d^3*
x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*((a^2 - b^2)*d^4*x^4 - 2*(a^2 - b^
2)*c^4)*cosh(d*x + c)^2 - 2*(4*(a^2 - b^2)*d^4*x^4 - 8*(a^2 - b^2)*c^4 - 3*(4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*
b^2*d*x - 3*b^2)*cosh(d*x + c)^2 + 24*(a*b*d^3*x^3 - 3*a*b*d^2*x^2 + 6*a*b*d*x - 6*a*b)*cosh(d*x + c))*sinh(d*
x + c)^2 + 3*b^2 - 16*(a*b*d^3*x^3 + 3*a*b*d^2*x^2 + 6*a*b*d*x + 6*a*b)*cosh(d*x + c) + 96*((a^2 - b^2)*d^2*x^
2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d^2*x^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d^2*x^2*sinh(d*x + c)^2)*d
ilog(-(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) + 96*((a^2 - b^2)*d^2*x^2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d^2*x^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)
*d^2*x^2*sinh(d*x + c)^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) - 32*((a^2 - b^2)*c^3*cosh(d*x + c)^2 + 2*(a^2 - b^2)*c^3*cosh(d*x + c)*sinh(d*x
+ c) + (a^2 - b^2)*c^3*sinh(d*x + c)^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) - 32*((a^2 - b^2)*c^3*cosh(d*x + c)^2 + 2*(a^2 - b^2)*c^3*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^
3*sinh(d*x + c)^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) + 32*(((a^2 -
b^2)*d^3*x^3 + (a^2 - b^2)*c^3)*cosh(d*x + c)^2 + 2*((a^2 - b^2)*d^3*x^3 + (a^2 - b^2)*c^3)*cosh(d*x + c)*sinh
(d*x + c) + ((a^2 - b^2)*d^3*x^3 + (a^2 - b^2)*c^3)*sinh(d*x + c)^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) + 32*(((a^2 - b^2)*d^3*x^3 + (a^2 - b^2)*c^3
)*cosh(d*x + c)^2 + 2*((a^2 - b^2)*d^3*x^3 + (a^2 - b^2)*c^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 - b^2)*d^3*x
^3 + (a^2 - b^2)*c^3)*sinh(d*x + c)^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) + 192*((a^2 - b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x
+ c) + (a^2 - b^2)*sinh(d*x + c)^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) + 192*((a^2 - b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x
+ c) + (a^2 - b^2)*sinh(d*x + c)^2)*polylog(4, -(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sin
h(d*x + c))*sqrt((a^2 - b^2)/b^2))/b) - 192*((a^2 - b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d*x*cosh(d*x + c)
*sinh(d*x + c) + (a^2 - b^2)*d*x*sinh(d*x + c)^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) - 192*((a^2 - b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d*x*
cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d*x*sinh(d*x + c)^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) - 4*(4*a*b*d^3*x^3 + 12*a*b*d^2*x^2 + 24*a*b*d
*x - (4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2)*cosh(d*x + c)^3 + 12*(a*b*d^3*x^3 - 3*a*b*d^2*x^2 + 6
*a*b*d*x - 6*a*b)*cosh(d*x + c)^2 + 24*a*b + 4*((a^2 - b^2)*d^4*x^4 - 2*(a^2 - b^2)*c^4)*cosh(d*x + c))*sinh(d
*x + c))/(b^3*d^4*cosh(d*x + c)^2 + 2*b^3*d^4*cosh(d*x + c)*sinh(d*x + c) + b^3*d^4*sinh(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \cosh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (8 \, {\left (a^{2} d^{4} e^{\left (2 \, c\right )} - b^{2} d^{4} e^{\left (2 \, c\right )}\right )} x^{4} + {\left (4 \, b^{2} d^{3} x^{3} e^{\left (4 \, c\right )} - 6 \, b^{2} d^{2} x^{2} e^{\left (4 \, c\right )} + 6 \, b^{2} d x e^{\left (4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 16 \, {\left (a b d^{3} x^{3} e^{\left (3 \, c\right )} - 3 \, a b d^{2} x^{2} e^{\left (3 \, c\right )} + 6 \, a b d x e^{\left (3 \, c\right )} - 6 \, a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 16 \, {\left (a b d^{3} x^{3} e^{c} + 3 \, a b d^{2} x^{2} e^{c} + 6 \, a b d x e^{c} + 6 \, a b e^{c}\right )} e^{\left (-d x\right )} + {\left (4 \, b^{2} d^{3} x^{3} + 6 \, b^{2} d^{2} x^{2} + 6 \, b^{2} d x + 3 \, b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{32 \, b^{3} d^{4}} - \frac {1}{8} \, \int \frac {16 \, {\left ({\left (a^{3} e^{c} - a b^{2} e^{c}\right )} x^{3} e^{\left (d x\right )} + {\left (a^{2} b - b^{3}\right )} x^{3}\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} + b^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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