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

Optimal. Leaf size=631 \[ \frac{i a^4 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac{i a^4 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2 \left (a^2+b^2\right )}-\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2 \left (a^2+b^2\right )}-\frac{i a^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2}+\frac{i a^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2}-\frac{a f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d \left (a^2+b^2\right )}+\frac{a^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac{2 a^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d \left (a^2+b^2\right )}+\frac{2 a^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{a (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d}+\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}-\frac{2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac{(e+f x) \sinh (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.957045, antiderivative size = 631, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {5581, 5449, 3296, 2638, 4180, 2279, 2391, 3718, 2190, 5567, 5573, 5561, 6742} \[ \frac{i a^4 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac{i a^4 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2 \left (a^2+b^2\right )}-\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2 \left (a^2+b^2\right )}-\frac{i a^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2}+\frac{i a^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2}-\frac{a f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d \left (a^2+b^2\right )}+\frac{a^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac{2 a^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d \left (a^2+b^2\right )}+\frac{2 a^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{a (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d}+\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}-\frac{2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac{(e+f x) \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5581

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(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*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n)/(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 5449

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

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 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]

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 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 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 6742

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

Rubi steps

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

Mathematica [A]  time = 4.78708, size = 481, normalized size = 0.76 \[ -\frac{\frac{a^3 \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac{1}{2} f (c+d x)^2\right )}{b^2 \left (a^2+b^2\right )}+\frac{-\frac{1}{2} a f \text{PolyLog}(2,\sinh (2 (c+d x))-\cosh (2 (c+d x)))-i b f \text{PolyLog}(2,-i (\sinh (c+d x)+\cosh (c+d x)))+i b f \text{PolyLog}(2,i (\sinh (c+d x)+\cosh (c+d x)))-a d e (c+d x)+a d e \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)+\frac{1}{2} a f (c+d x)^2+a c f (c+d x)+a f (c+d x) \log (2 \cosh (c+d x) (\cosh (c+d x)-\sinh (c+d x)))-a c f \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)+2 b d e \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))+2 b f (c+d x) \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-2 b c f \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))}{a^2+b^2}-\frac{d (e+f x) \sinh (c+d x)}{b}+\frac{f \cosh (c+d x)}{b}}{d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((f*Cosh[c + d*x])/b + (a^3*(-(f*(c + d*x)^2)/2 + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
+ f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e*Log[a + b*Sinh[c + d*x]] - c*f*Log[a + b*S
inh[c + d*x]] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))]))/(b^2*(a^2 + b^2)) + (-(a*d*e*(c + d*x)) + a*c*f*(c + d*x) + (a*f*(c + d*x)^2)/2 + 2*b*d*e*Ar
cTan[Cosh[c + d*x] + Sinh[c + d*x]] - 2*b*c*f*ArcTan[Cosh[c + d*x] + Sinh[c + d*x]] + 2*b*f*(c + d*x)*ArcTan[C
osh[c + d*x] + Sinh[c + d*x]] + a*f*(c + d*x)*Log[2*Cosh[c + d*x]*(Cosh[c + d*x] - Sinh[c + d*x])] + a*d*e*Log
[1 + Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]] - a*c*f*Log[1 + Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]] - I*b*f*Pol
yLog[2, (-I)*(Cosh[c + d*x] + Sinh[c + d*x])] + I*b*f*PolyLog[2, I*(Cosh[c + d*x] + Sinh[c + d*x])] - (a*f*Pol
yLog[2, -Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]])/2)/(a^2 + b^2) - (d*(e + f*x)*Sinh[c + d*x])/b)/d^2)

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Maple [B]  time = 0.309, size = 4066, normalized size = 6.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2/d^2*a^4*f/(a^2+b^2)^(3/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/b^2/d^2*a^4*f/(a
^2+b^2)^(3/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/b^2/d^2*a^3*f/(a^2+b^2)*dilog((b
*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/b^2/d^2*a^3*f/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(
1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/d^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*c+1/d*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)))*a*x+1/d^2*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)))*a*c+1/d*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)))*a*x+1/d^2*
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)))*a*c-2/d*f/(2*a^2+2*b^2)*ln(1+I*exp(d*
x+c))*a*x+4/d^2*f*c/(2*a^2+2*b^2)*b*arctan(exp(d*x+c))-1/d^2*f*c/(2*a^2+2*b^2)*a*ln(b*exp(2*d*x+2*c)+2*a*exp(d
*x+c)-b)+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))+2/d^2*f*c/(
2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))-2/d^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*a*c-2/d*f/(2*a^2+2*b^2)*ln(1-I*ex
p(d*x+c))*a*x+1/2*(d*f*x+d*e-f)/d^2/b*exp(d*x+c)-1/2*(d*f*x+d*e+f)/d^2/b*exp(-d*x-c)+1/2*a*f*x^2/b^2+2/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))-a*e*x/b^2-2*a/b^2/d^2*f*c*ln(
exp(d*x+c))+2*a/b^2/d*f*c*x+2/d*a^2*e/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/d^
2*a*f/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2*a^2*f/(a^2+b^2)^(3/2)*dilo
g((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2*a^2*f/(a^2+b^2)^(3/2)*dilog((-b*exp(d*x+c)+(a^2+
b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/2/d^2*a*f/(a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^
(1/2)))-1/2/d*a*e/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/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/d^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))*a+1/d^2*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)))*a+1/d^2*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)))*a+a/b^2/d^2*f*c^2+2*a/b^2/d*e*ln(exp(d*x+c))-2/b^2/d*a^4*f/(2*a^2+
2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-2/b^2/d^2*a^4*f/(2*a^2+2*b^2
)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/b^2/d^2*a^4*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))+2/b^2/d^2*a^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))+2/b^2/d*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*
x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+2/b^2/d^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c
)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/b^2/d^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)
+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/b^2/d^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a
^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/b^2/d^2*a^4*f*c/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2
+b^2)^(1/2))+1/b^2/d^2*a^4*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/b^2/
d*a^3*f/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/b^2/d^2*a^3*f/(a^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-1/b^2/d*a^4*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^
(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/b^2/d^2*a^4*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2
+b^2)^(1/2)))*c+2/d*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/
2)))*x+2/d^2*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-
2/d*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-2/d^2*a^2*f
/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/b^2/d*a^3*f/(a^2+b
^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/b^2/d^2*a^3*f/(a^2+b^2)*ln((-b*exp(d*x+c)+(
a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/b^2/d*a^4*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a
+(a^2+b^2)^(1/2)))*x-2*I*b/d*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-2*I*b/d^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))
*c+2*I*b/d*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x+2*I*b/d^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-2/d*e/(2*a^2+2*
b^2)*a*ln(1+exp(2*d*x+2*c))-4/d*e/(2*a^2+2*b^2)*b*arctan(exp(d*x+c))+1/d*e/(2*a^2+2*b^2)*a*ln(b*exp(2*d*x+2*c)
+2*a*exp(d*x+c)-b)-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))-2/d^2
*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))*a-2/b^2/d*a^2*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))-2/b^2/d*a^4*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))+1/b^2/d^2*a^3*f*c/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/b^2/d*a^4*e/(a^2+b^2)^(3/2)*arcta
nh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/b^2/d*a^3*e/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d
^2*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d^2*a^2*f
/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/2/d^2*a*f*c/(a^
2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2*a^2*f*c/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a
^2+b^2)^(1/2))-1/2/d*a*f/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/2/d^2*a*f/(a
^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/2/d*a*f/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2
+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/2/d^2*a*f/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(
1/2)))*c+1/d*a^2*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2*a^2*f/(a^2
+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d*a^2*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*
x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d^2*a^2*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-
a)/(-a+(a^2+b^2)^(1/2)))*c-2*I*b/d^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))+2*I*b/d^2*f/(2*a^2+2*b^2)*dilog(1+I
*exp(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, 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^{2} b^{2} + b^{4}\right )} d} - \frac{4 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \, a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \,{\left (d x + c\right )} a}{b^{2} d} - \frac{e^{\left (d x + c\right )}}{b d} + \frac{e^{\left (-d x - c\right )}}{b d}\right )} e - \frac{1}{4} \, f{\left (\frac{2 \,{\left (a d^{2} x^{2} e^{c} -{\left (b d x e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} +{\left (b d x + b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{b^{2} d^{2}} - \int -\frac{8 \,{\left (a^{4} x e^{\left (d x + c\right )} - a^{3} b x\right )}}{a^{2} b^{3} + b^{5} -{\left (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^{3} b^{2} e^{c} + a b^{4} e^{c}\right )} e^{\left (d x\right )}}\,{d x} + \int \frac{8 \,{\left (b x e^{\left (d x + c\right )} - a x\right )}}{a^{2} + b^{2} +{\left (a^{2} e^{\left (2 \, c\right )} + b^{2} 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)*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^4)*d) - 4*b*arctan(e^(-d*x - c))/((a
^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(
-d*x - c)/(b*d))*e - 1/4*f*(2*(a*d^2*x^2*e^c - (b*d*x*e^(2*c) - b*e^(2*c))*e^(d*x) + (b*d*x + b)*e^(-d*x))*e^(
-c)/(b^2*d^2) - integrate(-8*(a^4*x*e^(d*x + c) - a^3*b*x)/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^
(2*d*x) - 2*(a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) + integrate(8*(b*x*e^(d*x + c) - a*x)/(a^2 + b^2 + (a^2*e^(
2*c) + b^2*e^(2*c))*e^(2*d*x)), x))

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Fricas [B]  time = 3.11505, size = 3537, normalized size = 5.61 \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)*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - (a^2*b + b^3)*f)*co
sh(d*x + c)^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)^2 + (a^2*b + b^3)*f
- ((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^2)*c*d*e - 2*(a^3 + a*b^2)*c^2*f)*cosh(d*x
+ c) + 2*(a^3*f*cosh(d*x + c) + a^3*f*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) + a^3*f*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*b^2*f + 2*I*b^3*f)*cosh(d*x + c) + (2*a*b^2*f + 2*I*b^3*f)*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh
(d*x + c)) + ((2*a*b^2*f - 2*I*b^3*f)*cosh(d*x + c) + (2*a*b^2*f - 2*I*b^3*f)*sinh(d*x + c))*dilog(-I*cosh(d*x
+ c) - I*sinh(d*x + c)) + 2*((a^3*d*e - a^3*c*f)*cosh(d*x + c) + (a^3*d*e - a^3*c*f)*sinh(d*x + c))*log(2*b*c
osh(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)*cosh(d*x + c) + (
a^3*d*e - a^3*c*f)*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)*cosh(d*x + c) + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sin
h(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)*cosh
(d*x + c) + (a^3*d*f*x + a^3*c*f)*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*b^2*d*e + 2*I*b^3*d*e - 2*a*b^2*c*f - 2*I*b^3*c*f)*cosh
(d*x + c) + (2*a*b^2*d*e + 2*I*b^3*d*e - 2*a*b^2*c*f - 2*I*b^3*c*f)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*
x + c) + I) + ((2*a*b^2*d*e - 2*I*b^3*d*e - 2*a*b^2*c*f + 2*I*b^3*c*f)*cosh(d*x + c) + (2*a*b^2*d*e - 2*I*b^3*
d*e - 2*a*b^2*c*f + 2*I*b^3*c*f)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + ((2*a*b^2*d*f*x - 2*I
*b^3*d*f*x + 2*a*b^2*c*f - 2*I*b^3*c*f)*cosh(d*x + c) + (2*a*b^2*d*f*x - 2*I*b^3*d*f*x + 2*a*b^2*c*f - 2*I*b^3
*c*f)*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + ((2*a*b^2*d*f*x + 2*I*b^3*d*f*x + 2*a*b^2*c*
f + 2*I*b^3*c*f)*cosh(d*x + c) + (2*a*b^2*d*f*x + 2*I*b^3*d*f*x + 2*a*b^2*c*f + 2*I*b^3*c*f)*sinh(d*x + c))*lo
g(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^
2)*c*d*e - 2*(a^3 + a*b^2)*c^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)
)*sinh(d*x + c))/((a^2*b^2 + b^4)*d^2*cosh(d*x + c) + (a^2*b^2 + b^4)*d^2*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 ) \sinh ^{2}{\left (c + d x \right )} \tanh{\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)*sinh(d*x+c)**2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Timed out