\(\int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [28]

Optimal result

Integrand size = 26, antiderivative size = 400 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b f x}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {2 f \cosh (c+d x)}{3 a d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d} \]

[Out]

-1/4*b*f*x/a^2/d+1/2*b*(a^2+b^2)*(f*x+e)^2/a^4/f-2/3*f*cosh(d*x+c)/a/d^2-b^2*f*cosh(d*x+c)/a^3/d^2-1/9*f*cosh(
d*x+c)^3/a/d^2-b*(a^2+b^2)*(f*x+e)*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*(f*x+e)*ln(1+a*exp
(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*f*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^2-b*(a^2+b
^2)*f*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^2+2/3*(f*x+e)*sinh(d*x+c)/a/d+b^2*(f*x+e)*sinh(d*x+c)
/a^3/d+1/4*b*f*cosh(d*x+c)*sinh(d*x+c)/a^2/d^2+1/3*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/a/d-1/2*b*(f*x+e)*sinh(d*
x+c)^2/a^2/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5698, 3391, 3377, 2718, 5684, 5554, 2715, 8, 5680, 2221, 2317, 2438} \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \sinh (c+d x) \cosh (c+d x)}{4 a^2 d^2}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac {b f x}{4 a^2 d}-\frac {b f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^4 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {2 f \cosh (c+d x)}{3 a d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]

[In]

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

[Out]

-1/4*(b*f*x)/(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 8

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

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 2317

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 2438

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]

Rule 2715

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] && Integ
erQ[2*n]

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 5554

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 5680

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 5684

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 5698

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 5713

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 [A] (verified)

Time = 1.18 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.74 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {36 a^2 b c^2 f+36 b^3 c^2 f-36 a^2 b d^2 f x^2-36 b^3 d^2 f x^2+54 a^3 f \cosh (c+d x)+72 a b^2 f \cosh (c+d x)+18 a^2 b d f x \cosh (2 (c+d x))+2 a^3 f \cosh (3 (c+d x))+72 a^2 b c f \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 b^3 c f \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 a^2 b d f x \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 b^3 d f x \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 a^2 b c f \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 b^3 c f \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 a^2 b d f x \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 b^3 d f x \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-72 a^2 b c f \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )-72 b^3 c f \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )+72 a^2 b d e \log (b+a \sinh (c+d x))+72 b^3 d e \log (b+a \sinh (c+d x))+72 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+72 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-72 a^3 d e \sinh (c+d x)-72 a b^2 d e \sinh (c+d x)-54 a^3 d f x \sinh (c+d x)-72 a b^2 d f x \sinh (c+d x)+36 a^2 b d e \sinh ^2(c+d x)-24 a^3 d e \sinh ^3(c+d x)-9 a^2 b f \sinh (2 (c+d x))-6 a^3 d f x \sinh (3 (c+d x))}{72 a^4 d^2} \]

[In]

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

[Out]

-1/72*(36*a^2*b*c^2*f + 36*b^3*c^2*f - 36*a^2*b*d^2*f*x^2 - 36*b^3*d^2*f*x^2 + 54*a^3*f*Cosh[c + d*x] + 72*a*b
^2*f*Cosh[c + d*x] + 18*a^2*b*d*f*x*Cosh[2*(c + d*x)] + 2*a^3*f*Cosh[3*(c + d*x)] + 72*a^2*b*c*f*Log[1 + (a*E^
(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*b^3*c*f*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*a^2*b*d*f*x
*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*b^3*d*f*x*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])]
+ 72*a^2*b*c*f*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 72*b^3*c*f*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a
^2 + b^2])] + 72*a^2*b*d*f*x*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 72*b^3*d*f*x*Log[1 + (a*E^(c + d
*x))/(b + Sqrt[a^2 + b^2])] - 72*a^2*b*c*f*Log[a - 2*b*E^(c + d*x) - a*E^(2*(c + d*x))] - 72*b^3*c*f*Log[a - 2
*b*E^(c + d*x) - a*E^(2*(c + d*x))] + 72*a^2*b*d*e*Log[b + a*Sinh[c + d*x]] + 72*b^3*d*e*Log[b + a*Sinh[c + d*
x]] + 72*b*(a^2 + b^2)*f*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 72*b*(a^2 + b^2)*f*PolyLog[2, -(
(a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 72*a^3*d*e*Sinh[c + d*x] - 72*a*b^2*d*e*Sinh[c + d*x] - 54*a^3*d*f*x
*Sinh[c + d*x] - 72*a*b^2*d*f*x*Sinh[c + d*x] + 36*a^2*b*d*e*Sinh[c + d*x]^2 - 24*a^3*d*e*Sinh[c + d*x]^3 - 9*
a^2*b*f*Sinh[2*(c + d*x)] - 6*a^3*d*f*x*Sinh[3*(c + d*x)])/(a^4*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(372)=744\).

Time = 17.62 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1102\)

[In]

int((f*x+e)*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2465 vs. \(2 (370) = 740\).

Time = 0.31 (sec) , antiderivative size = 2465, normalized size of antiderivative = 6.16 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]

[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)

Sympy [F]

\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[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))

Giac [F]

\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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