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

Optimal. Leaf size=477 \[ \frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2}-\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3}+\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d}+\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d}-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.642494, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5565, 3296, 2637, 5446, 3310, 5561, 2190, 2531, 2282, 6589} \[ \frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2}-\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3}+\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d}+\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d}-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[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 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 5446

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

Rule 3310

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 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), 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 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 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)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac{a \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2}+\frac{\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac{\left (a^2+b^2\right ) \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{\left (a^2+b^2\right ) \int \frac{e^{c+d x} (e+f 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} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac{(2 a f) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d}-\frac{f \int (e+f x) \sinh ^2(c+d x) \, dx}{b d}\\ &=-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{\left (a^2+b^2\right ) (e+f 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 ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{f \int (e+f x) \, dx}{2 b d}-\frac{\left (2 \left (a^2+b^2\right ) f\right ) \int (e+f 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 ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^3 d}-\frac{\left (2 a f^2\right ) \int \cosh (c+d x) \, dx}{b^2 d^2}\\ &=\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{\left (a^2+b^2\right ) (e+f 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 ) (e+f 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 ) f (e+f 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 ) f (e+f 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 f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{\left (2 \left (a^2+b^2\right ) f^2\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 ) f^2\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{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{\left (a^2+b^2\right ) (e+f 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 ) (e+f 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 ) f (e+f 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 ) f (e+f 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 f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{\left (2 \left (a^2+b^2\right ) f^2\right ) \operatorname{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 ) f^2\right ) \operatorname{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{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{\left (a^2+b^2\right ) (e+f 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 ) (e+f 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 ) f (e+f 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 ) f (e+f 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 ) f^2 \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 ) f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}\\ \end{align*}

Mathematica [B]  time = 17.6291, size = 3021, normalized size = 6.33 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a^2 + b^2)*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a +
b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a +
b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (
3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c -
Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)
])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log
[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + S
qrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])
])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1
+ (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E
^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*
c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^
2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))
])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*Pol
yLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*b^3*(-1 + E^(2*c))) + Csch[c]*(Cos
h[2*c + 2*d*x]/(96*b^3*d^3) - Sinh[2*c + 2*d*x]/(96*b^3*d^3))*(-24*a*b*d^2*e^2*Cosh[d*x] - 48*a*b*d*e*f*Cosh[d
*x] - 48*a*b*f^2*Cosh[d*x] - 48*a*b*d^2*e*f*x*Cosh[d*x] - 48*a*b*d*f^2*x*Cosh[d*x] - 24*a*b*d^2*f^2*x^2*Cosh[d
*x] + 24*a*b*d^2*e^2*Cosh[2*c + d*x] + 48*a*b*d*e*f*Cosh[2*c + d*x] + 48*a*b*f^2*Cosh[2*c + d*x] + 48*a*b*d^2*
e*f*x*Cosh[2*c + d*x] + 48*a*b*d*f^2*x*Cosh[2*c + d*x] + 24*a*b*d^2*f^2*x^2*Cosh[2*c + d*x] + 48*a^2*d^3*e^2*x
*Cosh[c + 2*d*x] + 48*b^2*d^3*e^2*x*Cosh[c + 2*d*x] + 48*a^2*d^3*e*f*x^2*Cosh[c + 2*d*x] + 48*b^2*d^3*e*f*x^2*
Cosh[c + 2*d*x] + 16*a^2*d^3*f^2*x^3*Cosh[c + 2*d*x] + 16*b^2*d^3*f^2*x^3*Cosh[c + 2*d*x] + 48*a^2*d^3*e^2*x*C
osh[3*c + 2*d*x] + 48*b^2*d^3*e^2*x*Cosh[3*c + 2*d*x] + 48*a^2*d^3*e*f*x^2*Cosh[3*c + 2*d*x] + 48*b^2*d^3*e*f*
x^2*Cosh[3*c + 2*d*x] + 16*a^2*d^3*f^2*x^3*Cosh[3*c + 2*d*x] + 16*b^2*d^3*f^2*x^3*Cosh[3*c + 2*d*x] + 24*a*b*d
^2*e^2*Cosh[2*c + 3*d*x] - 48*a*b*d*e*f*Cosh[2*c + 3*d*x] + 48*a*b*f^2*Cosh[2*c + 3*d*x] + 48*a*b*d^2*e*f*x*Co
sh[2*c + 3*d*x] - 48*a*b*d*f^2*x*Cosh[2*c + 3*d*x] + 24*a*b*d^2*f^2*x^2*Cosh[2*c + 3*d*x] - 24*a*b*d^2*e^2*Cos
h[4*c + 3*d*x] + 48*a*b*d*e*f*Cosh[4*c + 3*d*x] - 48*a*b*f^2*Cosh[4*c + 3*d*x] - 48*a*b*d^2*e*f*x*Cosh[4*c + 3
*d*x] + 48*a*b*d*f^2*x*Cosh[4*c + 3*d*x] - 24*a*b*d^2*f^2*x^2*Cosh[4*c + 3*d*x] - 6*b^2*d^2*e^2*Cosh[3*c + 4*d
*x] + 6*b^2*d*e*f*Cosh[3*c + 4*d*x] - 3*b^2*f^2*Cosh[3*c + 4*d*x] - 12*b^2*d^2*e*f*x*Cosh[3*c + 4*d*x] + 6*b^2
*d*f^2*x*Cosh[3*c + 4*d*x] - 6*b^2*d^2*f^2*x^2*Cosh[3*c + 4*d*x] + 6*b^2*d^2*e^2*Cosh[5*c + 4*d*x] - 6*b^2*d*e
*f*Cosh[5*c + 4*d*x] + 3*b^2*f^2*Cosh[5*c + 4*d*x] + 12*b^2*d^2*e*f*x*Cosh[5*c + 4*d*x] - 6*b^2*d*f^2*x*Cosh[5
*c + 4*d*x] + 6*b^2*d^2*f^2*x^2*Cosh[5*c + 4*d*x] + 12*b^2*d^2*e^2*Sinh[c] + 12*b^2*d*e*f*Sinh[c] + 6*b^2*f^2*
Sinh[c] + 24*b^2*d^2*e*f*x*Sinh[c] + 12*b^2*d*f^2*x*Sinh[c] + 12*b^2*d^2*f^2*x^2*Sinh[c] - 24*a*b*d^2*e^2*Sinh
[d*x] - 48*a*b*d*e*f*Sinh[d*x] - 48*a*b*f^2*Sinh[d*x] - 48*a*b*d^2*e*f*x*Sinh[d*x] - 48*a*b*d*f^2*x*Sinh[d*x]
- 24*a*b*d^2*f^2*x^2*Sinh[d*x] + 24*a*b*d^2*e^2*Sinh[2*c + d*x] + 48*a*b*d*e*f*Sinh[2*c + d*x] + 48*a*b*f^2*Si
nh[2*c + d*x] + 48*a*b*d^2*e*f*x*Sinh[2*c + d*x] + 48*a*b*d*f^2*x*Sinh[2*c + d*x] + 24*a*b*d^2*f^2*x^2*Sinh[2*
c + d*x] + 48*a^2*d^3*e^2*x*Sinh[c + 2*d*x] + 48*b^2*d^3*e^2*x*Sinh[c + 2*d*x] + 48*a^2*d^3*e*f*x^2*Sinh[c + 2
*d*x] + 48*b^2*d^3*e*f*x^2*Sinh[c + 2*d*x] + 16*a^2*d^3*f^2*x^3*Sinh[c + 2*d*x] + 16*b^2*d^3*f^2*x^3*Sinh[c +
2*d*x] + 48*a^2*d^3*e^2*x*Sinh[3*c + 2*d*x] + 48*b^2*d^3*e^2*x*Sinh[3*c + 2*d*x] + 48*a^2*d^3*e*f*x^2*Sinh[3*c
+ 2*d*x] + 48*b^2*d^3*e*f*x^2*Sinh[3*c + 2*d*x] + 16*a^2*d^3*f^2*x^3*Sinh[3*c + 2*d*x] + 16*b^2*d^3*f^2*x^3*S
inh[3*c + 2*d*x] + 24*a*b*d^2*e^2*Sinh[2*c + 3*d*x] - 48*a*b*d*e*f*Sinh[2*c + 3*d*x] + 48*a*b*f^2*Sinh[2*c + 3
*d*x] + 48*a*b*d^2*e*f*x*Sinh[2*c + 3*d*x] - 48*a*b*d*f^2*x*Sinh[2*c + 3*d*x] + 24*a*b*d^2*f^2*x^2*Sinh[2*c +
3*d*x] - 24*a*b*d^2*e^2*Sinh[4*c + 3*d*x] + 48*a*b*d*e*f*Sinh[4*c + 3*d*x] - 48*a*b*f^2*Sinh[4*c + 3*d*x] - 48
*a*b*d^2*e*f*x*Sinh[4*c + 3*d*x] + 48*a*b*d*f^2*x*Sinh[4*c + 3*d*x] - 24*a*b*d^2*f^2*x^2*Sinh[4*c + 3*d*x] - 6
*b^2*d^2*e^2*Sinh[3*c + 4*d*x] + 6*b^2*d*e*f*Sinh[3*c + 4*d*x] - 3*b^2*f^2*Sinh[3*c + 4*d*x] - 12*b^2*d^2*e*f*
x*Sinh[3*c + 4*d*x] + 6*b^2*d*f^2*x*Sinh[3*c + 4*d*x] - 6*b^2*d^2*f^2*x^2*Sinh[3*c + 4*d*x] + 6*b^2*d^2*e^2*Si
nh[5*c + 4*d*x] - 6*b^2*d*e*f*Sinh[5*c + 4*d*x] + 3*b^2*f^2*Sinh[5*c + 4*d*x] + 12*b^2*d^2*e*f*x*Sinh[5*c + 4*
d*x] - 6*b^2*d*f^2*x*Sinh[5*c + 4*d*x] + 6*b^2*d^2*f^2*x^2*Sinh[5*c + 4*d*x])

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Maple [F]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{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)^2*cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, e^{2}{\left (\frac{{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} - \frac{8 \,{\left (a^{2} + b^{2}\right )}{\left (d x + c\right )}}{b^{3} d} - \frac{4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d} - \frac{8 \,{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d}\right )} + \frac{{\left (16 \,{\left (a^{2} d^{3} f^{2} e^{\left (2 \, c\right )} + b^{2} d^{3} f^{2} e^{\left (2 \, c\right )}\right )} x^{3} + 48 \,{\left (a^{2} d^{3} e f e^{\left (2 \, c\right )} + b^{2} d^{3} e f e^{\left (2 \, c\right )}\right )} x^{2} + 3 \,{\left (2 \, b^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \,{\left (2 \, d^{2} e f - d f^{2}\right )} b^{2} x e^{\left (4 \, c\right )} -{\left (2 \, d e f - f^{2}\right )} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 24 \,{\left (a b d^{2} f^{2} x^{2} e^{\left (3 \, c\right )} + 2 \,{\left (d^{2} e f - d f^{2}\right )} a b x e^{\left (3 \, c\right )} - 2 \,{\left (d e f - f^{2}\right )} a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} + 24 \,{\left (a b d^{2} f^{2} x^{2} e^{c} + 2 \,{\left (d^{2} e f + d f^{2}\right )} a b x e^{c} + 2 \,{\left (d e f + f^{2}\right )} a b e^{c}\right )} e^{\left (-d x\right )} + 3 \,{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 2 \,{\left (2 \, d^{2} e f + d f^{2}\right )} b^{2} x +{\left (2 \, d e f + f^{2}\right )} b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{48 \, b^{3} d^{3}} - \int -\frac{2 \,{\left ({\left (a^{2} b f^{2} + b^{3} f^{2}\right )} x^{2} + 2 \,{\left (a^{2} b e f + b^{3} e f\right )} x -{\left ({\left (a^{3} f^{2} e^{c} + a b^{2} f^{2} e^{c}\right )} x^{2} + 2 \,{\left (a^{3} e f e^{c} + a b^{2} e f e^{c}\right )} x\right )} e^{\left (d x\right )}\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^2*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 8*(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) - 8*(a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d)) + 1/48
*(16*(a^2*d^3*f^2*e^(2*c) + b^2*d^3*f^2*e^(2*c))*x^3 + 48*(a^2*d^3*e*f*e^(2*c) + b^2*d^3*e*f*e^(2*c))*x^2 + 3*
(2*b^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*b^2*x*e^(4*c) - (2*d*e*f - f^2)*b^2*e^(4*c))*e^(2*d*x) - 24
*(a*b*d^2*f^2*x^2*e^(3*c) + 2*(d^2*e*f - d*f^2)*a*b*x*e^(3*c) - 2*(d*e*f - f^2)*a*b*e^(3*c))*e^(d*x) + 24*(a*b
*d^2*f^2*x^2*e^c + 2*(d^2*e*f + d*f^2)*a*b*x*e^c + 2*(d*e*f + f^2)*a*b*e^c)*e^(-d*x) + 3*(2*b^2*d^2*f^2*x^2 +
2*(2*d^2*e*f + d*f^2)*b^2*x + (2*d*e*f + f^2)*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^3) - integrate(-2*((a^2*b*f^2 +
b^3*f^2)*x^2 + 2*(a^2*b*e*f + b^3*e*f)*x - ((a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*f*e^
c)*x)*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 [C]  time = 2.64666, size = 6356, normalized size = 13.32 \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)^2*cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(6*b^2*d^2*f^2*x^2 + 6*b^2*d^2*e^2 + 6*b^2*d*e*f + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b
^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^4 + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f
+ b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*b^2*f^2 - 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 - 12*(2*a*b*d^2*f^2*x^2 + 2*a*b*d^2
*e^2 - 4*a*b*d*e*f + 4*a*b*f^2 + 4*(a*b*d^2*e*f - a*b*d*f^2)*x - (2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*
e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*((a^2 + b^2)*d^3*f^2*x^3
+ 3*(a^2 + b^2)*d^3*e*f*x^2 + 3*(a^2 + b^2)*d^3*e^2*x + 6*(a^2 + b^2)*c*d^2*e^2 - 6*(a^2 + b^2)*c^2*d*e*f + 2*
(a^2 + b^2)*c^3*f^2)*cosh(d*x + c)^2 - 2*(8*(a^2 + b^2)*d^3*f^2*x^3 + 24*(a^2 + b^2)*d^3*e*f*x^2 + 24*(a^2 + b
^2)*d^3*e^2*x + 48*(a^2 + b^2)*c*d^2*e^2 - 48*(a^2 + b^2)*c^2*d*e*f + 16*(a^2 + b^2)*c^3*f^2 - 9*(2*b^2*d^2*f^
2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^2 + 36*(a*b*d^2
*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)
^2 + 6*(2*b^2*d^2*e*f + b^2*d*f^2)*x + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 + 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^
2*e*f + a*b*d*f^2)*x)*cosh(d*x + c) + 96*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)^2 + 2*((a^2
+ b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*si
nh(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*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d*f^2*x + (
a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*sinh(d*x + c)^2)*dil
og((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)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^2
*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d^2*e^2 - 2*(a^
2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^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*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c
)^2 + 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^
2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*s
inh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*(((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(
a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f
*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(
a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^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*f^2*
x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*
d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x +
c) + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^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)*f^2*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^
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)*f^2*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^
2 + b^2)*f^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*f^2*x^2 + 6*a*b*d^2*e^2 + 12*a*b*d*e*f + 12*a*b*f^2 + 3*(2*b^2*d^
2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^3 - 18*(a*b
*d^2*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 + 12*(a*
b*d^2*e*f + a*b*d*f^2)*x - 8*((a^2 + b^2)*d^3*f^2*x^3 + 3*(a^2 + b^2)*d^3*e*f*x^2 + 3*(a^2 + b^2)*d^3*e^2*x +
6*(a^2 + b^2)*c*d^2*e^2 - 6*(a^2 + b^2)*c^2*d*e*f + 2*(a^2 + b^2)*c^3*f^2)*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(-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)**2*cosh(d*x+c)**3/(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 )}^{2} \cosh \left (d x + c\right )^{3}}{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)^2*cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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