3.3.35 \(\int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\) [235]

Optimal. Leaf size=432 \[ \frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d} \]

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5685, 3377, 2718, 5480, 3391, 30, 5681, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 \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 {2 \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 {2 x \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 {2 x \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^2 \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^2 \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^3 \left (a^2-b^2\right )}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{4 b d^3}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}+\frac {x^2}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[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 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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 5480

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 5681

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 5685

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 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]

Rubi steps

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

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Mathematica [A]
time = 2.19, size = 697, normalized size = 1.61 \begin {gather*} \frac {\frac {8 \left (-a^2+b^2\right ) \left (2 d^3 e^{2 c} x^3-3 d^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 )-3 d^2 e^{2 c} 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 )-3 d^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 )-3 d^2 e^{2 c} 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 )-6 d \left (1+e^{2 c}\right ) x \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 d \left (1+e^{2 c}\right ) x \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 \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 e^{2 c} \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 \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 e^{2 c} \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 )\right )}{d^3 \left (1+e^{2 c}\right )}-\frac {24 a b \cosh (d x) \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac {3 b^2 \cosh (2 d x) \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 b^2 \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^3}+8 \left (a^2-b^2\right ) x^3 \tanh (c)}{24 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*(-a^2 + b^2)*(2*d^3*E^(2*c)*x^3 - 3*d^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 - b^2)*E^(2*c)])]
- 3*d^2*E^(2*c)*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 - b^2)*E^(2*c)])] - 3*d^2*x^2*Log[1 + (b*E^(
2*c + d*x))/(a*E^c + Sqrt[(a^2 - b^2)*E^(2*c)])] - 3*d^2*E^(2*c)*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(
a^2 - b^2)*E^(2*c)])] - 6*d*(1 + E^(2*c))*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 - b^2)*E^(2*c)])
)] - 6*d*(1 + E^(2*c))*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 - b^2)*E^(2*c)]))] + 6*PolyLog[3, -
((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 - b^2)*E^(2*c)]))] + 6*E^(2*c)*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 - b^2)*E^(2*c)]))] + 6*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 - b^2)*E^(2*c)]))] + 6*E^(2*
c)*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 - b^2)*E^(2*c)]))]))/(d^3*(1 + E^(2*c))) - (24*a*b*Cosh[d
*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*b^2*Cosh[2*d*x]*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[
2*c]))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*(-2*d*x*Cosh[2*c] + (1 +
2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^3 + 8*(a^2 - b^2)*x^3*Tanh[c])/(24*b^3)

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Maple [F]
time = 0.79, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \cosh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(16*(a^2*d^3*e^(2*c) - b^2*d^3*e^(2*c))*x^3 + 3*(2*b^2*d^2*x^2*e^(4*c) - 2*b^2*d*x*e^(4*c) + b^2*e^(4*c))
*e^(2*d*x) - 24*(a*b*d^2*x^2*e^(3*c) - 2*a*b*d*x*e^(3*c) + 2*a*b*e^(3*c))*e^(d*x) - 24*(a*b*d^2*x^2*e^c + 2*a*
b*d*x*e^c + 2*a*b*e^c)*e^(-d*x) + 3*(2*b^2*d^2*x^2 + 2*b^2*d*x + b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^3) - 1/8*int
egrate(16*((a^3*e^c - a*b^2*e^c)*x^2*e^(d*x) + (a^2*b - b^3)*x^2)/(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. 1622 vs. \(2 (402) = 804\).
time = 0.50, size = 1622, normalized size = 3.75 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(6*b^2*d^2*x^2 + 3*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c)^4 + 3*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2
)*sinh(d*x + c)^4 + 6*b^2*d*x - 24*(a*b*d^2*x^2 - 2*a*b*d*x + 2*a*b)*cosh(d*x + c)^3 - 12*(2*a*b*d^2*x^2 - 4*a
*b*d*x + 4*a*b - (2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*((a^2 - b^2)*d^3*x^3 +
2*(a^2 - b^2)*c^3)*cosh(d*x + c)^2 - 2*(8*(a^2 - b^2)*d^3*x^3 + 16*(a^2 - b^2)*c^3 - 9*(2*b^2*d^2*x^2 - 2*b^2*
d*x + b^2)*cosh(d*x + c)^2 + 36*(a*b*d^2*x^2 - 2*a*b*d*x + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*b^2 - 24*
(a*b*d^2*x^2 + 2*a*b*d*x + 2*a*b)*cosh(d*x + c) + 96*((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)*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) + 96*((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)*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) + 48*((a^2 - b^2)*c^2*cosh(d*x +
c)^2 + 2*(a^2 - b^2)*c^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^2*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) + 48*((a^2 - b^2)*c^2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*
c^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^2*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) + 48*(((a^2 - b^2)*d^2*x^2 - (a^2 - b^2)*c^2)*cosh(d*x + c)^2 + 2*((a^2 - b^
2)*d^2*x^2 - (a^2 - b^2)*c^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 - b^2)*d^2*x^2 - (a^2 - b^2)*c^2)*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) + 48*(((a^2 - b^2)*d^2*x^2 - (a^2 - b^2)*c^2)*cosh(d*x + c)^2 + 2*((a^2 - b^2)*d^2*x^2 - (a^2 - b^2)*c^2)*
cosh(d*x + c)*sinh(d*x + c) + ((a^2 - b^2)*d^2*x^2 - (a^2 - b^2)*c^2)*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) - 96*((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(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) - 96*((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(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*(6*a*b*d^2*x^2 + 12*a*
b*d*x - 3*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c)^3 + 18*(a*b*d^2*x^2 - 2*a*b*d*x + 2*a*b)*cosh(d*x +
c)^2 + 12*a*b + 8*((a^2 - b^2)*d^3*x^3 + 2*(a^2 - b^2)*c^3)*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^3*cosh(d*x +
c)^2 + 2*b^3*d^3*cosh(d*x + c)*sinh(d*x + c) + b^3*d^3*sinh(d*x + c)^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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