3.28 \(\int \frac{(e+f x) \cosh ^3(c+d x)}{a+b \text{csch}(c+d x)} \, dx\)

Optimal. Leaf size=400 \[ -\frac{b f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{b f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^4 d^2}-\frac{b^2 f \cosh (c+d x)}{a^3 d^2}-\frac{b \left (a^2+b^2\right ) (e+f x) \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x) \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^4 d}+\frac{b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}+\frac{b f \sinh (c+d x) \cosh (c+d x)}{4 a^2 d^2}-\frac{b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac{b f x}{4 a^2 d}-\frac{f \cosh ^3(c+d x)}{9 a d^2}-\frac{2 f \cosh (c+d x)}{3 a d^2}+\frac{2 (e+f x) \sinh (c+d x)}{3 a d}+\frac{(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.551043, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5594, 5579, 3310, 3296, 2638, 5565, 5446, 2635, 8, 5561, 2190, 2279, 2391} \[ -\frac{b f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{b f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^4 d^2}-\frac{b^2 f \cosh (c+d x)}{a^3 d^2}-\frac{b \left (a^2+b^2\right ) (e+f x) \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x) \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^4 d}+\frac{b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}+\frac{b f \sinh (c+d x) \cosh (c+d x)}{4 a^2 d^2}-\frac{b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac{b f x}{4 a^2 d}-\frac{f \cosh ^3(c+d x)}{9 a d^2}-\frac{2 f \cosh (c+d x)}{3 a d^2}+\frac{2 (e+f x) \sinh (c+d x)}{3 a d}+\frac{(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5594

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

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 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 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 2638

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

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 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 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 8

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

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

Mathematica [C]  time = 5.82828, size = 743, normalized size = 1.86 \[ -\frac{\text{csch}(c+d x) (a \sinh (c+d x)+b) \left (f \left (36 b \left (a^2+2 b^2\right ) \left (\text{PolyLog}\left (2,\frac{a e^{c+d x}}{\sqrt{a^2+b^2}-b}\right )+\text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )+(c+d x) \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )+(c+d x) \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )-c \log (a \sinh (c+d x)+b)-\frac{1}{2} (c+d x)^2\right )-18 a d x \left (a^2+4 b^2\right ) \sinh (c+d x)+18 a \left (a^2+4 b^2\right ) \cosh (c+d x)-9 a^2 b \sinh (2 (c+d x))+18 a^2 b d x \cosh (2 (c+d x))-6 a^3 d x \sinh (3 (c+d x))+2 a^3 \cosh (3 (c+d x))\right )+\frac{9}{2} a^2 f \left (8 b \left (\text{PolyLog}\left (2,\frac{\left (b-\sqrt{a^2+b^2}\right ) e^{c+d x}}{a}\right )+\text{PolyLog}\left (2,\frac{\left (\sqrt{a^2+b^2}+b\right ) e^{c+d x}}{a}\right )\right )+4 b \log \left (\frac{\left (\sqrt{a^2+b^2}-b\right ) e^{c+d x}}{a}+1\right ) \left (4 i \sin ^{-1}\left (\frac{\sqrt{1+\frac{i b}{a}}}{\sqrt{2}}\right )+2 c+2 d x+i \pi \right )+4 b \log \left (1-\frac{\left (\sqrt{a^2+b^2}+b\right ) e^{c+d x}}{a}\right ) \left (-4 i \sin ^{-1}\left (\frac{\sqrt{1+\frac{i b}{a}}}{\sqrt{2}}\right )+2 c+2 d x+i \pi \right )-32 b \sin ^{-1}\left (\frac{\sqrt{1+\frac{i b}{a}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{(b+i a) \cot \left (\frac{1}{4} (2 i c+2 i d x+\pi )\right )}{\sqrt{a^2+b^2}}\right )-4 i \pi b \log (a \sinh (c+d x)+b)-8 b c \log \left (\frac{a \sinh (c+d x)}{b}+1\right )-8 a d x \sinh (c+d x)+8 a \cosh (c+d x)-b (2 c+2 d x+i \pi )^2\right )+12 d e \left (-3 a \left (a^2+2 b^2\right ) \sinh (c+d x)+3 b \left (a^2+2 b^2\right ) \log (a \sinh (c+d x)+b)+3 a^2 b \sinh ^2(c+d x)-2 a^3 \sinh ^3(c+d x)\right )-36 a^2 d e (a \sinh (c+d x)-b \log (a \sinh (c+d x)+b))\right )}{72 a^4 d^2 (a+b \text{csch}(c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-36*a^2*d*e*(-(b*Log[b + a*Sinh[c + d*x]]) + a*Sinh[c + d*x]) + (9*a^2*
f*(-(b*(2*c + I*Pi + 2*d*x)^2) - 32*b*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*ArcTan[((I*a + b)*Cot[((2*I)*c + Pi +
(2*I)*d*x)/4])/Sqrt[a^2 + b^2]] + 8*a*Cosh[c + d*x] + 4*b*(2*c + I*Pi + 2*d*x + (4*I)*ArcSin[Sqrt[1 + (I*b)/a]
/Sqrt[2]])*Log[1 + ((-b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] + 4*b*(2*c + I*Pi + 2*d*x - (4*I)*ArcSin[Sqrt[1 + (
I*b)/a]/Sqrt[2]])*Log[1 - ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] - (4*I)*b*Pi*Log[b + a*Sinh[c + d*x]] - 8*b*c
*Log[1 + (a*Sinh[c + d*x])/b] + 8*b*(PolyLog[2, ((b - Sqrt[a^2 + b^2])*E^(c + d*x))/a] + PolyLog[2, ((b + Sqrt
[a^2 + b^2])*E^(c + d*x))/a]) - 8*a*d*x*Sinh[c + d*x]))/2 + 12*d*e*(3*b*(a^2 + 2*b^2)*Log[b + a*Sinh[c + d*x]]
- 3*a*(a^2 + 2*b^2)*Sinh[c + d*x] + 3*a^2*b*Sinh[c + d*x]^2 - 2*a^3*Sinh[c + d*x]^3) + f*(18*a*(a^2 + 4*b^2)*
Cosh[c + d*x] + 18*a^2*b*d*x*Cosh[2*(c + d*x)] + 2*a^3*Cosh[3*(c + d*x)] + 36*b*(a^2 + 2*b^2)*(-(c + d*x)^2/2
+ (c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 +
b^2])] - c*Log[b + a*Sinh[c + d*x]] + PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + PolyLog[2, -((a*E^
(c + d*x))/(b + Sqrt[a^2 + b^2]))]) - 18*a*(a^2 + 4*b^2)*d*x*Sinh[c + d*x] - 9*a^2*b*Sinh[2*(c + d*x)] - 6*a^3
*d*x*Sinh[3*(c + d*x)])))/(72*a^4*d^2*(a + b*Csch[c + d*x]))

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Maple [B]  time = 0.191, size = 1102, 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)^3/(a+b*csch(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{24} \, e{\left (\frac{{\left (3 \, a b e^{\left (-d x - c\right )} - a^{2} - 3 \,{\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{a^{3} d} + \frac{24 \,{\left (a^{2} b + b^{3}\right )}{\left (d x + c\right )}}{a^{4} d} + \frac{3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + a^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \,{\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-d x - c\right )}}{a^{3} d} + \frac{24 \,{\left (a^{2} b + b^{3}\right )} \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{4} d}\right )} - \frac{1}{144} \, f{\left (\frac{{\left (72 \,{\left (a^{2} b d^{2} e^{\left (3 \, c\right )} + b^{3} d^{2} e^{\left (3 \, c\right )}\right )} x^{2} - 2 \,{\left (3 \, a^{3} d x e^{\left (6 \, c\right )} - a^{3} e^{\left (6 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 9 \,{\left (2 \, a^{2} b d x e^{\left (5 \, c\right )} - a^{2} b e^{\left (5 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 18 \,{\left (3 \, a^{3} e^{\left (4 \, c\right )} + 4 \, a b^{2} e^{\left (4 \, c\right )} -{\left (3 \, a^{3} d e^{\left (4 \, c\right )} + 4 \, a b^{2} d e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (d x\right )} + 18 \,{\left (3 \, a^{3} e^{\left (2 \, c\right )} + 4 \, a b^{2} e^{\left (2 \, c\right )} +{\left (3 \, a^{3} d e^{\left (2 \, c\right )} + 4 \, a b^{2} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (-d x\right )} + 9 \,{\left (2 \, a^{2} b d x e^{c} + a^{2} b e^{c}\right )} e^{\left (-2 \, d x\right )} + 2 \,{\left (3 \, a^{3} d x + a^{3}\right )} e^{\left (-3 \, d x\right )}\right )} e^{\left (-3 \, c\right )}}{a^{4} d^{2}} - 18 \, \int \frac{16 \,{\left ({\left (a^{2} b^{2} e^{c} + b^{4} e^{c}\right )} x e^{\left (d x\right )} -{\left (a^{3} b + a b^{3}\right )} x\right )}}{a^{5} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b e^{\left (d x + c\right )} - a^{5}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*e*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b +
b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^
3*d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4*d)) - 1/144*f*((72*(a^2*b*d^2*e^(
3*c) + b^3*d^2*e^(3*c))*x^2 - 2*(3*a^3*d*x*e^(6*c) - a^3*e^(6*c))*e^(3*d*x) + 9*(2*a^2*b*d*x*e^(5*c) - a^2*b*e
^(5*c))*e^(2*d*x) + 18*(3*a^3*e^(4*c) + 4*a*b^2*e^(4*c) - (3*a^3*d*e^(4*c) + 4*a*b^2*d*e^(4*c))*x)*e^(d*x) + 1
8*(3*a^3*e^(2*c) + 4*a*b^2*e^(2*c) + (3*a^3*d*e^(2*c) + 4*a*b^2*d*e^(2*c))*x)*e^(-d*x) + 9*(2*a^2*b*d*x*e^c +
a^2*b*e^c)*e^(-2*d*x) + 2*(3*a^3*d*x + a^3)*e^(-3*d*x))*e^(-3*c)/(a^4*d^2) - 18*integrate(16*((a^2*b^2*e^c + b
^4*e^c)*x*e^(d*x) - (a^3*b + a*b^3)*x)/(a^5*e^(2*d*x + 2*c) + 2*a^4*b*e^(d*x + c) - a^5), x))

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Fricas [B]  time = 1.98321, size = 5828, normalized size = 14.57 \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)^3/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

1/144*(2*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^6 + 2*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*sinh(d*x + c)
^6 - 6*a^3*d*f*x - 9*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^5 - 3*(6*a^2*b*d*f*x + 6*a^2*b*d*e
- 3*a^2*b*f - 4*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*a^3*d*e + 18*((3*a^3 + 4*
a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c)^4 + 3*(6*(3*a^3 + 4*a*b^2)*d*f*x + 6
*(3*a^3 + 4*a*b^2)*d*e + 10*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^2 - 6*(3*a^3 + 4*a*b^2)*f - 15*(2*
a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*a^3*f + 72*((a^2*b + b^3)*d^2*f*x^2 +
2*(a^2*b + b^3)*d^2*e*x + 4*(a^2*b + b^3)*c*d*e - 2*(a^2*b + b^3)*c^2*f)*cosh(d*x + c)^3 + 2*(36*(a^2*b + b^3)
*d^2*f*x^2 + 72*(a^2*b + b^3)*d^2*e*x + 144*(a^2*b + b^3)*c*d*e - 72*(a^2*b + b^3)*c^2*f + 20*(3*a^3*d*f*x + 3
*a^3*d*e - a^3*f)*cosh(d*x + c)^3 - 45*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^2 + 36*((3*a^3 +
4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 18*((3*a^3 + 4*
a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e + (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c)^2 + 6*(5*(3*a^3*d*f*x + 3*a^3*d*e -
a^3*f)*cosh(d*x + c)^4 - 3*(3*a^3 + 4*a*b^2)*d*f*x - 15*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^
3 - 3*(3*a^3 + 4*a*b^2)*d*e + 18*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3*a^3 + 4*a*b^2)*f)*cosh(
d*x + c)^2 - 3*(3*a^3 + 4*a*b^2)*f + 36*((a^2*b + b^3)*d^2*f*x^2 + 2*(a^2*b + b^3)*d^2*e*x + 4*(a^2*b + b^3)*c
*d*e - 2*(a^2*b + b^3)*c^2*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 9*(2*a^2*b*d*f*x + 2*a^2*b*d*e + a^2*b*f)*cosh(
d*x + c) - 144*((a^2*b + b^3)*f*cosh(d*x + c)^3 + 3*(a^2*b + b^3)*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b +
b^3)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b + b^3)*f*sinh(d*x + c)^3)*dilog((b*cosh(d*x + c) + b*sinh(d*x +
c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 144*((a^2*b + b^3)*f*cosh(d*x +
c)^3 + 3*(a^2*b + b^3)*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b + b^3)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (a^
2*b + b^3)*f*sinh(d*x + c)^3)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*s
qrt((a^2 + b^2)/a^2) - a)/a + 1) - 144*(((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*cosh(d*x + c)^3 + 3*((a^2*b +
b^3)*d*e - (a^2*b + b^3)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*cosh(d
*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*sinh(d*x + c)^3)*log(2*a*cosh(d*x + c) + 2*a
*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) - 144*(((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*cosh(d*x + c)
^3 + 3*((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*e - (a^2*b +
b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*e - (a^2*b + b^3)*c*f)*sinh(d*x + c)^3)*log(2*a*co
sh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) - 144*(((a^2*b + b^3)*d*f*x + (a^2*b + b^3)
*c*f)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*si
nh(d*x + c)^3)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/
a^2) - a)/a) - 144*(((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d*f*x + (a^2*
b + b^3)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)*sinh(d
*x + c)^2 + ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*sinh(d*x + c)^3)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c)
- (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) - 3*(6*a^2*b*d*f*x - 4*(3*a^3*d*f*x + 3*a
^3*d*e - a^3*f)*cosh(d*x + c)^5 + 6*a^2*b*d*e + 15*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^4 + 3
*a^2*b*f - 24*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c)^3 - 72*((a
^2*b + b^3)*d^2*f*x^2 + 2*(a^2*b + b^3)*d^2*e*x + 4*(a^2*b + b^3)*c*d*e - 2*(a^2*b + b^3)*c^2*f)*cosh(d*x + c)
^2 + 12*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e + (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c))*sinh(d*x + c))/
(a^4*d^2*cosh(d*x + c)^3 + 3*a^4*d^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^4*d^2*cosh(d*x + c)*sinh(d*x + c)^2 +
a^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)**3/(a+b*csch(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 )^{3}}{b \operatorname{csch}\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)^3/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

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