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

Optimal. Leaf size=894 \[ \text{result too large to display} \]

[Out]

(a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*d) + ((e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)*ArcTan[
E^(c + d*x)])/(b*(a^2 + b^2)^2*d) - (a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (a^3*(e + f*x)*L
og[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - ((I/2)*a^
2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) - ((I/2)*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (I*a^4*f*PolyLo
g[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) + ((I/2)*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^
2) + ((I/2)*a^2*f*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) + ((I/2)*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^4*
f*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) - ((I/2)*a^4*f*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*
d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^3*f*PolyLog[2, -(
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 +
b^2)^2*d^2) + (a^2*f*Sech[c + d*x])/(2*b^3*d^2) - (f*Sech[c + d*x])/(2*b*d^2) - (a^4*f*Sech[c + d*x])/(2*b^3*(
a^2 + b^2)*d^2) + (a*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*d) - (a^3*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*(a^2 + b^2)
*d) - (a*f*Tanh[c + d*x])/(2*b^2*d^2) + (a^3*f*Tanh[c + d*x])/(2*b^2*(a^2 + b^2)*d^2) + (a^2*(e + f*x)*Sech[c
+ d*x]*Tanh[c + d*x])/(2*b^3*d) - ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^4*(e + f*x)*Sech[c + d*
x]*Tanh[c + d*x])/(2*b^3*(a^2 + b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 1.41524, antiderivative size = 894, normalized size of antiderivative = 1., number of steps used = 55, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {5567, 5455, 4180, 2279, 2391, 4185, 5583, 5451, 3767, 8, 5573, 5561, 2190, 6742, 3718} \[ -\frac{(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}-\frac{2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d^2}-\frac{f \text{sech}(c+d x) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac{(e+f x) \text{sech}(c+d x) \tanh (c+d x) a^4}{2 b^3 \left (a^2+b^2\right ) d}-\frac{(e+f x) \text{sech}^2(c+d x) a^3}{2 b^2 \left (a^2+b^2\right ) d}-\frac{(e+f x) \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac{(e+f x) \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3}{\left (a^2+b^2\right )^2 d}+\frac{(e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac{f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}-\frac{f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac{f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3}{2 \left (a^2+b^2\right )^2 d^2}+\frac{f \tanh (c+d x) a^3}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac{(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b^3 d}-\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{2 b^3 d^2}+\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right ) a^2}{2 b^3 d^2}+\frac{f \text{sech}(c+d x) a^2}{2 b^3 d^2}+\frac{(e+f x) \text{sech}(c+d x) \tanh (c+d x) a^2}{2 b^3 d}+\frac{(e+f x) \text{sech}^2(c+d x) a}{2 b^2 d}-\frac{f \tanh (c+d x) a}{2 b^2 d^2}+\frac{(e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac{f \text{sech}(c+d x)}{2 b d^2}-\frac{(e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*d) + ((e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)*ArcTan[
E^(c + d*x)])/(b*(a^2 + b^2)^2*d) - (a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (a^3*(e + f*x)*L
og[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - ((I/2)*a^
2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) - ((I/2)*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (I*a^4*f*PolyLo
g[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) + ((I/2)*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^
2) + ((I/2)*a^2*f*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) + ((I/2)*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^4*
f*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) - ((I/2)*a^4*f*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*
d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^3*f*PolyLog[2, -(
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 +
b^2)^2*d^2) + (a^2*f*Sech[c + d*x])/(2*b^3*d^2) - (f*Sech[c + d*x])/(2*b*d^2) - (a^4*f*Sech[c + d*x])/(2*b^3*(
a^2 + b^2)*d^2) + (a*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*d) - (a^3*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*(a^2 + b^2)
*d) - (a*f*Tanh[c + d*x])/(2*b^2*d^2) + (a^3*f*Tanh[c + d*x])/(2*b^2*(a^2 + b^2)*d^2) + (a^2*(e + f*x)*Sech[c
+ d*x]*Tanh[c + d*x])/(2*b^3*d) - ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^4*(e + f*x)*Sech[c + d*
x]*Tanh[c + d*x])/(2*b^3*(a^2 + b^2)*d)

Rule 5567

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sec
h[c + d*x]*Tanh[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]

Rule 5455

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]*Tanh[(a_.) + (b_.)*(x_)]^(p_), x_Symbol] :> Int[(c + d
*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 5583

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1),
x], x] - Dist[a/b, Int[((e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[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 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 3767

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

Rule 8

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

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rubi steps

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

Mathematica [A]  time = 7.86361, size = 588, normalized size = 0.66 \[ \frac{-2 a^3 f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-2 a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-i b f \left (3 a^2+b^2\right ) \text{PolyLog}\left (2,-i e^{c+d x}\right )+i b f \left (3 a^2+b^2\right ) \text{PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )+d \left (a^2+b^2\right ) (e+f x) \text{sech}^2(c+d x) (a-b \sinh (c+d x))-2 a^3 f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )-2 a^3 f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-f \left (a^2+b^2\right ) \text{sech}(c+d x) (a \sinh (c+d x)+b)+6 a^2 b d e \tan ^{-1}\left (e^{c+d x}\right )-2 a^3 d e \log (a+b \sinh (c+d x))+3 i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-6 a^2 b c f \tan ^{-1}\left (e^{c+d x}\right )+2 a^3 c f \log (a+b \sinh (c+d x))-2 a^3 d e (c+d x)+2 a^3 d e \log \left (e^{2 (c+d x)}+1\right )+2 a^3 c f (c+d x)-2 a^3 c f \log \left (e^{2 (c+d x)}+1\right )+2 a^3 f (c+d x) \log \left (e^{2 (c+d x)}+1\right )+2 b^3 d e \tan ^{-1}\left (e^{c+d x}\right )+i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )-2 b^3 c f \tan ^{-1}\left (e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*d*e*(c + d*x) + 2*a^3*c*f*(c + d*x) + 6*a^2*b*d*e*ArcTan[E^(c + d*x)] + 2*b^3*d*e*ArcTan[E^(c + d*x)]
- 6*a^2*b*c*f*ArcTan[E^(c + d*x)] - 2*b^3*c*f*ArcTan[E^(c + d*x)] + (3*I)*a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d
*x)] + I*b^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (3*I)*a^2*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] - I*b^3*f*(c
+ d*x)*Log[1 + I*E^(c + d*x)] - 2*a^3*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*a^3*f*(c
+ d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*a^3*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^3*c*f*Log[1 +
E^(2*(c + d*x))] + 2*a^3*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] - 2*a^3*d*e*Log[a + b*Sinh[c + d*x]] + 2*a^3*c*
f*Log[a + b*Sinh[c + d*x]] - I*b*(3*a^2 + b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*(3*a^2 + b^2)*f*PolyLog[2,
I*E^(c + d*x)] - 2*a^3*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*a^3*f*PolyLog[2, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))] + a^3*f*PolyLog[2, -E^(2*(c + d*x))] - (a^2 + b^2)*f*Sech[c + d*x]*(b + a*Sinh[c
+ d*x]) + (a^2 + b^2)*d*(e + f*x)*Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)^2*d^2)

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Maple [B]  time = 0.202, size = 2284, normalized size = 2.6 \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)*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

3/d^2/(a^2+b^2)^(3/2)*b^2*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2+2/d*e/(2*a^2
+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2-2/(a^2+b^2)^(3/2)/d*a^4*e/(2*a^2
+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d^2*f*c/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*
(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2-1/d/(a^2+b^2)^(3/2)*b^4*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+
2*a)/(a^2+b^2)^(1/2))+1/d/(a^2+b^2)^(1/2)*b^2*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2)
)-2/(a^2+b^2)/d*a^3*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-2/(a^2+b^2)/d
^2*a^3*f/(2*a^2+2*b^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*a^3*f/(2*a^2
+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*ln((b*e
xp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/(a^2+b^2)/d*a^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x+2/(
a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-2/(a^2+b^2)/d^2*a^3*f*c/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c
))+2/(a^2+b^2)/d^2*a^3*f*c/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/(a^2+b^2)/d*a^3*f/(2*a^2+2*b^
2)*ln(1-I*exp(d*x+c))*x+2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c-3*I*b/(a^2+b^2)/d*a^2*f/(2*a^
2+2*b^2)*ln(1+I*exp(d*x+c))*x-3*I*b/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c+3*I/(a^2+b^2)/d*a^2
*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*b*x+3*I/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*b*c-I*b^3/(a^
2+b^2)/d*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-I*b^3/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c+I*b^3/(
a^2+b^2)/d*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x+I*b^3/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c-6*b/(
a^2+b^2)/d^2*a^2*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2*b^3/(a^2+b^2)/d^2*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))
+I*b^3/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))+6*b/(a^2+b^2)/d*a^2*e/(2*a^2+2*b^2)*arctan(exp(d*x+
c))-I*b^3/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+3*I/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*dilog(1-I*
exp(d*x+c))*b-3*I*b/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+1/d^2/(a^2+b^2)^(3/2)*b^4*f*c/(2*a
^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-3/d/(a^2+b^2)^(3/2)*b^2*e/(2*a^2+2*b^2)*arctanh(1/
2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2-1/d^2/(a^2+b^2)^(1/2)*b^2*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d
*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(a^2+b^2)^(3/2)/d^2*a^4*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+
b^2)^(1/2))+(-b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+b*d*
f*x*exp(d*x+c)-b*f*exp(3*d*x+3*c)+a*f*exp(2*d*x+2*c)+b*d*e*exp(d*x+c)-f*b*exp(d*x+c)+a*f)/d^2/(a^2+b^2)/(1+exp
(2*d*x+2*c))^2+2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+2*b^3/(a^2+b^2)/d*e/(2*a^2+2*b^2)*arc
tan(exp(d*x+c))+2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))-2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*
dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*dilog((-b*exp(
d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+2/(a^2+b^2)/d*a^3*e/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-2/(a^2+
b^2)/d*a^3*e/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - a^3*log(e^(-2*d*x - 2*c) +
1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (3*a^2*b + b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (b*e^(-d*
x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)
*e^(-4*d*x - 4*c))*d))*e - f*(((b*d*x*e^(3*c) + b*e^(3*c))*e^(3*d*x) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x)
- (b*d*x*e^c - b*e^c)*e^(d*x) - a)/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^(4*d*x) + 2*(a^
2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x)) - integrate(-2*(a^4*x*e^(d*x + c) - a^3*b*x)/(a^4*b + 2*a^2*b^3 +
b^5 - (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 2*a^3*b^2*e^c + a*b^4*e^c)*e^
(d*x)), x) - integrate(-(2*a^3*x - (3*a^2*b*e^c + b^3*e^c)*x*e^(d*x))/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(2*c) +
2*a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x))

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Fricas [B]  time = 4.00183, size = 11386, normalized size = 12.74 \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)*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c)^3 + 2*((a^2*b + b^3)*d*f*x +
(a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*sinh(d*x + c)^3 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3
+ a*b^2)*f)*cosh(d*x + c)^2 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*f - 3*((a^2*b + b
^3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(a^3 + a*b^2)*f - 2*((a^2*
b + b^3)*d*f*x + (a^2*b + b^3)*d*e - (a^2*b + b^3)*f)*cosh(d*x + c) + 2*(a^3*f*cosh(d*x + c)^4 + 4*a^3*f*cosh(
d*x + c)*sinh(d*x + c)^3 + a^3*f*sinh(d*x + c)^4 + 2*a^3*f*cosh(d*x + c)^2 + a^3*f + 2*(3*a^3*f*cosh(d*x + c)^
2 + a^3*f)*sinh(d*x + c)^2 + 4*(a^3*f*cosh(d*x + c)^3 + a^3*f*cosh(d*x + c))*sinh(d*x + c))*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) + 2*(a^3*f*cosh
(d*x + c)^4 + 4*a^3*f*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*f*sinh(d*x + c)^4 + 2*a^3*f*cosh(d*x + c)^2 + a^3*f
+ 2*(3*a^3*f*cosh(d*x + c)^2 + a^3*f)*sinh(d*x + c)^2 + 4*(a^3*f*cosh(d*x + c)^3 + a^3*f*cosh(d*x + c))*sinh(d
*x + c))*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) - ((2*a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^4 + (8*a^3*f + 4*I*(3*a^2*b + b^3)*f)*cosh(d*x +
c)*sinh(d*x + c)^3 + (2*a^3*f + I*(3*a^2*b + b^3)*f)*sinh(d*x + c)^4 + 2*a^3*f + (4*a^3*f + 2*I*(3*a^2*b + b^3
)*f)*cosh(d*x + c)^2 + (4*a^3*f + (12*a^3*f + 6*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 2*I*(3*a^2*b + b^3)*f)*
sinh(d*x + c)^2 + I*(3*a^2*b + b^3)*f + ((8*a^3*f + 4*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^3 + (8*a^3*f + 4*I*(3
*a^2*b + b^3)*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - ((2*a^3*f - I*(3*a^2
*b + b^3)*f)*cosh(d*x + c)^4 + (8*a^3*f - 4*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^3*f - I*
(3*a^2*b + b^3)*f)*sinh(d*x + c)^4 + 2*a^3*f + (4*a^3*f - 2*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + (4*a^3*f +
(12*a^3*f - 6*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 - 2*I*(3*a^2*b + b^3)*f)*sinh(d*x + c)^2 - I*(3*a^2*b + b^3
)*f + ((8*a^3*f - 4*I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^3 + (8*a^3*f - 4*I*(3*a^2*b + b^3)*f)*cosh(d*x + c))*si
nh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 2*(a^3*d*e - a^3*c*f + (a^3*d*e - a^3*c*f)*cosh(d*x +
c)^4 + 4*(a^3*d*e - a^3*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*d*e - a^3*c*f)*sinh(d*x + c)^4 + 2*(a^3*d*e
- a^3*c*f)*cosh(d*x + c)^2 + 2*(a^3*d*e - a^3*c*f + 3*(a^3*d*e - a^3*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
4*((a^3*d*e - a^3*c*f)*cosh(d*x + c)^3 + (a^3*d*e - a^3*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x +
c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(a^3*d*e - a^3*c*f + (a^3*d*e - a^3*c*f)*cosh(d*
x + c)^4 + 4*(a^3*d*e - a^3*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*d*e - a^3*c*f)*sinh(d*x + c)^4 + 2*(a^3*
d*e - a^3*c*f)*cosh(d*x + c)^2 + 2*(a^3*d*e - a^3*c*f + 3*(a^3*d*e - a^3*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 4*((a^3*d*e - a^3*c*f)*cosh(d*x + c)^3 + (a^3*d*e - a^3*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x
+ c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(a^3*d*f*x + a^3*c*f + (a^3*d*f*x + a^3*c*f)*
cosh(d*x + c)^4 + 4*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c)^
4 + 2*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^2 + 2*(a^3*d*f*x + a^3*c*f + 3*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 + 4*((a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^3 + (a^3*d*f*x + a^3*c*f)*cosh(d*x + c))*sinh(d*x +
c))*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) + 2*(a^3*d*f*x + a^3*c*f + (a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^4 + 4*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)*si
nh(d*x + c)^3 + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c)^4 + 2*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^2 + 2*(a^3*d*f*x
+ a^3*c*f + 3*(a^3*d*f*x + a^3*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^3*d*f*x + a^3*c*f)*cosh(d*x + c)
^3 + (a^3*d*f*x + a^3*c*f)*cosh(d*x + c))*sinh(d*x + c))*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) - (2*a^3*d*e - 2*a^3*c*f + (2*a^3*d*e - 2*a^3*c*f + I*(
3*a^2*b + b^3)*d*e - I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^4 + (8*a^3*d*e - 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*e
- 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^3*d*e - 2*a^3*c*f + I*(3*a^2*b + b^3)*d*e - I
*(3*a^2*b + b^3)*c*f)*sinh(d*x + c)^4 + I*(3*a^2*b + b^3)*d*e - I*(3*a^2*b + b^3)*c*f + (4*a^3*d*e - 4*a^3*c*f
+ 2*I*(3*a^2*b + b^3)*d*e - 2*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^2 + (4*a^3*d*e - 4*a^3*c*f + 2*I*(3*a^2*b
+ b^3)*d*e - 2*I*(3*a^2*b + b^3)*c*f + (12*a^3*d*e - 12*a^3*c*f + 6*I*(3*a^2*b + b^3)*d*e - 6*I*(3*a^2*b + b^3
)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^3*d*e - 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*e - 4*I*(3*a^2*b + b
^3)*c*f)*cosh(d*x + c)^3 + (8*a^3*d*e - 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*e - 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*
x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) - (2*a^3*d*e - 2*a^3*c*f + (2*a^3*d*e - 2*a^3*c*
f - I*(3*a^2*b + b^3)*d*e + I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^4 + (8*a^3*d*e - 8*a^3*c*f - 4*I*(3*a^2*b + b
^3)*d*e + 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^3*d*e - 2*a^3*c*f - I*(3*a^2*b + b^3)*
d*e + I*(3*a^2*b + b^3)*c*f)*sinh(d*x + c)^4 - I*(3*a^2*b + b^3)*d*e + I*(3*a^2*b + b^3)*c*f + (4*a^3*d*e - 4*
a^3*c*f - 2*I*(3*a^2*b + b^3)*d*e + 2*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^2 + (4*a^3*d*e - 4*a^3*c*f - 2*I*(3
*a^2*b + b^3)*d*e + 2*I*(3*a^2*b + b^3)*c*f + (12*a^3*d*e - 12*a^3*c*f - 6*I*(3*a^2*b + b^3)*d*e + 6*I*(3*a^2*
b + b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^3*d*e - 8*a^3*c*f - 4*I*(3*a^2*b + b^3)*d*e + 4*I*(3*a^
2*b + b^3)*c*f)*cosh(d*x + c)^3 + (8*a^3*d*e - 8*a^3*c*f - 4*I*(3*a^2*b + b^3)*d*e + 4*I*(3*a^2*b + b^3)*c*f)*
cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) - (2*a^3*d*f*x + 2*a^3*c*f + (2*a^3*d*f*x
+ 2*a^3*c*f - I*(3*a^2*b + b^3)*d*f*x - I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^4 + (8*a^3*d*f*x + 8*a^3*c*f - 4
*I*(3*a^2*b + b^3)*d*f*x - 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^3*d*f*x + 2*a^3*c*f -
I*(3*a^2*b + b^3)*d*f*x - I*(3*a^2*b + b^3)*c*f)*sinh(d*x + c)^4 - I*(3*a^2*b + b^3)*d*f*x - I*(3*a^2*b + b^3
)*c*f + (4*a^3*d*f*x + 4*a^3*c*f - 2*I*(3*a^2*b + b^3)*d*f*x - 2*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^2 + (4*a
^3*d*f*x + 4*a^3*c*f - 2*I*(3*a^2*b + b^3)*d*f*x - 2*I*(3*a^2*b + b^3)*c*f + (12*a^3*d*f*x + 12*a^3*c*f - 6*I*
(3*a^2*b + b^3)*d*f*x - 6*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^3*d*f*x + 8*a^3*c*f
- 4*I*(3*a^2*b + b^3)*d*f*x - 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^3 + (8*a^3*d*f*x + 8*a^3*c*f - 4*I*(3*a^2
*b + b^3)*d*f*x - 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x + c)
+ 1) - (2*a^3*d*f*x + 2*a^3*c*f + (2*a^3*d*f*x + 2*a^3*c*f + I*(3*a^2*b + b^3)*d*f*x + I*(3*a^2*b + b^3)*c*f)
*cosh(d*x + c)^4 + (8*a^3*d*f*x + 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*f*x + 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x +
c)*sinh(d*x + c)^3 + (2*a^3*d*f*x + 2*a^3*c*f + I*(3*a^2*b + b^3)*d*f*x + I*(3*a^2*b + b^3)*c*f)*sinh(d*x + c)
^4 + I*(3*a^2*b + b^3)*d*f*x + I*(3*a^2*b + b^3)*c*f + (4*a^3*d*f*x + 4*a^3*c*f + 2*I*(3*a^2*b + b^3)*d*f*x +
2*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^2 + (4*a^3*d*f*x + 4*a^3*c*f + 2*I*(3*a^2*b + b^3)*d*f*x + 2*I*(3*a^2*b
+ b^3)*c*f + (12*a^3*d*f*x + 12*a^3*c*f + 6*I*(3*a^2*b + b^3)*d*f*x + 6*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c)^
2)*sinh(d*x + c)^2 + ((8*a^3*d*f*x + 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*f*x + 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x
+ c)^3 + (8*a^3*d*f*x + 8*a^3*c*f + 4*I*(3*a^2*b + b^3)*d*f*x + 4*I*(3*a^2*b + b^3)*c*f)*cosh(d*x + c))*sinh(
d*x + c))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - 3*((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c)^2 - (a^2*b + b^3)*f + 2*(2*(a^3 + a*b^2)*d*
f*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*f)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d
*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^2*sinh(d*x
+ c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2 + 2*(3*(a^4 + 2*a^2*b^2
+ b^4)*d^2*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)*sinh(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2 + b^4)*d^2*cos
h(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right ) \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x)*tanh(c + d*x)**3/(a + b*sinh(c + d*x)), x)

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Giac [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)*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out