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

Optimal. Leaf size=403 \[ \frac{a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}+\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}-\frac{a^3 e x}{b^4}-\frac{a^3 f x^2}{2 b^4}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}-\frac{a (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac{a e x}{2 b^2}-\frac{a f x^2}{4 b^2}-\frac{f \sinh ^3(c+d x)}{9 b d^2}-\frac{f \sinh (c+d x)}{3 b d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d} \]

[Out]

-((a^3*e*x)/b^4) - (a*e*x)/(2*b^2) - (a^3*f*x^2)/(2*b^4) - (a*f*x^2)/(4*b^2) + (a^2*(e + f*x)*Cosh[c + d*x])/(
b^3*d) + (a*f*Cosh[c + d*x]^2)/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f
*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b
^2]))])/(b^4*d^2) - (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (
a^2*f*Sinh[c + d*x])/(b^3*d^2) - (f*Sinh[c + d*x])/(3*b*d^2) - (a*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^
2*d) - (f*Sinh[c + d*x]^3)/(9*b*d^2)

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Rubi [A]  time = 0.690013, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {5579, 5447, 2633, 3310, 5565, 3296, 2637, 3322, 2264, 2190, 2279, 2391} \[ \frac{a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}+\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}-\frac{a^3 e x}{b^4}-\frac{a^3 f x^2}{2 b^4}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}-\frac{a (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac{a e x}{2 b^2}-\frac{a f x^2}{4 b^2}-\frac{f \sinh ^3(c+d x)}{9 b d^2}-\frac{f \sinh (c+d x)}{3 b d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*e*x)/b^4) - (a*e*x)/(2*b^2) - (a^3*f*x^2)/(2*b^4) - (a*f*x^2)/(4*b^2) + (a^2*(e + f*x)*Cosh[c + d*x])/(
b^3*d) + (a*f*Cosh[c + d*x]^2)/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f
*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b
^2]))])/(b^4*d^2) - (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (
a^2*f*Sinh[c + d*x])/(b^3*d^2) - (f*Sinh[c + d*x])/(3*b*d^2) - (a*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^
2*d) - (f*Sinh[c + d*x]^3)/(9*b*d^2)

Rule 5579

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*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] && IGtQ[p, 0]

Rule 5447

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac{a \int (e+f x) \cosh ^2(c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac{f \int \cosh ^3(c+d x) \, dx}{3 b d}\\ &=\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{a^3 \int (e+f x) \, dx}{b^4}+\frac{a^2 \int (e+f x) \sinh (c+d x) \, dx}{b^3}-\frac{a \int (e+f x) \, dx}{2 b^2}+\frac{\left (a^2 \left (a^2+b^2\right )\right ) \int \frac{e+f x}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac{(i f) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{3 b d^2}\\ &=-\frac{a^3 e x}{b^4}-\frac{a e x}{2 b^2}-\frac{a^3 f x^2}{2 b^4}-\frac{a f x^2}{4 b^2}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac{f \sinh (c+d x)}{3 b d^2}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{f \sinh ^3(c+d x)}{9 b d^2}+\frac{\left (2 a^2 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^4}-\frac{\left (a^2 f\right ) \int \cosh (c+d x) \, dx}{b^3 d}\\ &=-\frac{a^3 e x}{b^4}-\frac{a e x}{2 b^2}-\frac{a^3 f x^2}{2 b^4}-\frac{a f x^2}{4 b^2}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}-\frac{f \sinh (c+d x)}{3 b d^2}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{f \sinh ^3(c+d x)}{9 b d^2}+\frac{\left (2 a^2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3}-\frac{\left (2 a^2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3}\\ &=-\frac{a^3 e x}{b^4}-\frac{a e x}{2 b^2}-\frac{a^3 f x^2}{2 b^4}-\frac{a f x^2}{4 b^2}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}-\frac{f \sinh (c+d x)}{3 b d^2}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{f \sinh ^3(c+d x)}{9 b d^2}-\frac{\left (a^2 \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}+\frac{\left (a^2 \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}\\ &=-\frac{a^3 e x}{b^4}-\frac{a e x}{2 b^2}-\frac{a^3 f x^2}{2 b^4}-\frac{a f x^2}{4 b^2}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}-\frac{f \sinh (c+d x)}{3 b d^2}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{f \sinh ^3(c+d x)}{9 b d^2}-\frac{\left (a^2 \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}+\frac{\left (a^2 \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}\\ &=-\frac{a^3 e x}{b^4}-\frac{a e x}{2 b^2}-\frac{a^3 f x^2}{2 b^4}-\frac{a f x^2}{4 b^2}+\frac{a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac{a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{a^2 \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a^2 \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a^2 f \sinh (c+d x)}{b^3 d^2}-\frac{f \sinh (c+d x)}{3 b d^2}-\frac{a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{f \sinh ^3(c+d x)}{9 b d^2}\\ \end{align*}

Mathematica [A]  time = 3.05828, size = 676, normalized size = 1.68 \[ -\frac{-72 a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}-a}\right )+72 a^2 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}+a}\right )+144 a^2 d e \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt{a^2+b^2}}\right )-72 a^2 c f \sqrt{a^2+b^2} \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{a-\sqrt{a^2+b^2}}+1\right )-72 a^2 d f x \sqrt{a^2+b^2} \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{a-\sqrt{a^2+b^2}}+1\right )+72 a^2 c f \sqrt{a^2+b^2} \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}+a}+1\right )+72 a^2 d f x \sqrt{a^2+b^2} \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}+a}+1\right )-144 a^2 c f \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt{a^2+b^2}}\right )-72 a^2 b d e \cosh (c+d x)+72 a^2 b f \sinh (c+d x)-72 a^2 b d f x \cosh (c+d x)-36 a^3 c^2 f+72 a^3 c d e+72 a^3 d^2 e x+36 a^3 d^2 f x^2-18 a b^2 c^2 f+18 a b^2 d e \sinh (2 (c+d x))+36 a b^2 c d e+18 a b^2 d f x \sinh (2 (c+d x))-9 a b^2 f \cosh (2 (c+d x))+36 a b^2 d^2 e x+18 a b^2 d^2 f x^2-18 b^3 d e \cosh (c+d x)-6 b^3 d e \cosh (3 (c+d x))+18 b^3 f \sinh (c+d x)+2 b^3 f \sinh (3 (c+d x))-18 b^3 d f x \cosh (c+d x)-6 b^3 d f x \cosh (3 (c+d x))}{72 b^4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(72*a^3*c*d*e + 36*a*b^2*c*d*e - 36*a^3*c^2*f - 18*a*b^2*c^2*f + 72*a^3*d^2*e*x + 36*a*b^2*d^2*e*x + 36*a^3*d
^2*f*x^2 + 18*a*b^2*d^2*f*x^2 + 144*a^2*Sqrt[a^2 + b^2]*d*e*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sq
rt[a^2 + b^2]] - 144*a^2*Sqrt[a^2 + b^2]*c*f*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]]
- 72*a^2*b*d*e*Cosh[c + d*x] - 18*b^3*d*e*Cosh[c + d*x] - 72*a^2*b*d*f*x*Cosh[c + d*x] - 18*b^3*d*f*x*Cosh[c +
d*x] - 9*a*b^2*f*Cosh[2*(c + d*x)] - 6*b^3*d*e*Cosh[3*(c + d*x)] - 6*b^3*d*f*x*Cosh[3*(c + d*x)] - 72*a^2*Sqr
t[a^2 + b^2]*c*f*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - 72*a^2*Sqrt[a^2 + b^2]*d
*f*x*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] + 72*a^2*Sqrt[a^2 + b^2]*c*f*Log[1 + (
b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 72*a^2*Sqrt[a^2 + b^2]*d*f*x*Log[1 + (b*(Cosh[c +
d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] - 72*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c
+ d*x]))/(-a + Sqrt[a^2 + b^2])] + 72*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/
(a + Sqrt[a^2 + b^2]))] + 72*a^2*b*f*Sinh[c + d*x] + 18*b^3*f*Sinh[c + d*x] + 18*a*b^2*d*e*Sinh[2*(c + d*x)] +
18*a*b^2*d*f*x*Sinh[2*(c + d*x)] + 2*b^3*f*Sinh[3*(c + d*x)])/(72*b^4*d^2)

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Maple [B]  time = 0.106, size = 1128, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*a*(2*d*f*x+2*d*e-f)/b^2/d^2*exp(2*d*x+2*c)+1/16*a*(2*d*f*x+2*d*e+f)/b^2/d^2*exp(-2*d*x-2*c)-1/2*a^3*f*x^
2/b^4-1/4*a*f*x^2/b^2-a^4/b^4/d^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2))
)-2*a^4/b^4/d*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+a^4/b^4/d^2*f/(a^2+b^2)^(1/2
)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/8*(4*a^2+b^2)*(d*f*x+d*e+f)/b^3/d^2*exp(-d*x
-c)-a^3*e*x/b^4-1/2*a*e*x/b^2-2*a^2/b^2/d*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+
a^2/b^2/d^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-a^2/b^2/d^2*f/(a^2
+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+a^4/b^4/d*f/(a^2+b^2)^(1/2)*ln((-b*exp
(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+a^4/b^4/d^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(
1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-a^4/b^4/d*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(
1/2)))*x-a^4/b^4/d^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2*a^4/b^4/d^
2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+a^2/b^2/d*f/(a^2+b^2)^(1/2)*ln((-b*exp
(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+a^2/b^2/d^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(
1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-a^2/b^2/d*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(
1/2)))*x-a^2/b^2/d^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2*a^2/b^2/d^
2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/72*(3*d*f*x+3*d*e+f)/d^2/b*exp(-3*d*
x-3*c)+1/72*(3*d*f*x+3*d*e-f)/d^2/b*exp(3*d*x+3*c)+1/8*(4*a^2*d*f*x+b^2*d*f*x+4*a^2*d*e+b^2*d*e-4*a^2*f-b^2*f)
/b^3/d^2*exp(d*x+c)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.80926, size = 5268, normalized size = 13.07 \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)*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/144*(2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*sinh(d*x + c)
^6 + 6*b^3*d*f*x - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*e
- 3*a*b^2*f - 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*b^3*d*e + 18*((4*a^2*b +
b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^4 + 3*(6*(4*a^2*b + b^3)*d*f*x + 6*(4*a^2*
b + b^3)*d*e + 10*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^2 - 6*(4*a^2*b + b^3)*f - 15*(2*a*b^2*d*f*x
+ 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*b^3*f - 36*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 +
a*b^2)*d^2*e*x)*cosh(d*x + c)^3 - 2*(18*(2*a^3 + a*b^2)*d^2*f*x^2 + 36*(2*a^3 + a*b^2)*d^2*e*x - 20*(3*b^3*d*
f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^3 + 45*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^2 - 36*((4
*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^3 + 18*((4*a^2*b +
b^3)*d*f*x + (4*a^2*b + b^3)*d*e + (4*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f
)*cosh(d*x + c)^4 + 3*(4*a^2*b + b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^3 + 3*(
4*a^2*b + b^3)*d*e + 18*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 3*
(4*a^2*b + b^3)*f - 18*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 + a*b^2)*d^2*e*x)*cosh(d*x + c))*sinh(d*x + c)^2
+ 144*(a^2*b*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*b*f*cosh(d*x + c)*sinh(d*x +
c)^2 + a^2*b*f*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 144*(a^2*b*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh(d*x +
c)^2*sinh(d*x + c) + 3*a^2*b*f*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*b*f*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)
*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b +
1) - 144*((a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) +
3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*e - a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2
+ b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 144*((a^2*b*d*e - a
^2*b*c*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*e - a^2*b*c*f
)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*e - a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(
d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 144*((a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)
^3 + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)*sin
h(d*x + c)^2 + (a^2*b*d*f*x + a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh
(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 144*((a^2*b*d*f*x + a^2*b*c*f)
*cosh(d*x + c)^3 + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*f*x + a^2*b*c*f)*cos
h(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*f*x + a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x
+ c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 9*(2*a*b^2*d*f*x
+ 2*a*b^2*d*e + a*b^2*f)*cosh(d*x + c) + 3*(6*a*b^2*d*f*x + 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)
^5 + 6*a*b^2*d*e - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^4 + 3*a*b^2*f + 24*((4*a^2*b + b^3
)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^3 - 36*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3
+ a*b^2)*d^2*e*x)*cosh(d*x + c)^2 + 12*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e + (4*a^2*b + b^3)*f)*cosh(
d*x + c))*sinh(d*x + c))/(b^4*d^2*cosh(d*x + c)^3 + 3*b^4*d^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d^2*cosh(d
*x + c)*sinh(d*x + c)^2 + b^4*d^2*sinh(d*x + c)^3)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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