3.5.17 \(\int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [417]

Optimal. Leaf size=894 \[ \frac {a^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \text {ArcTan}\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 {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^4 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {i a^4 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^4 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {i a^4 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \text {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-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} \]

[Out]

-2*a^4*(f*x+e)*arctan(exp(d*x+c))/b/(a^2+b^2)^2/d-1/2*a^4*f*sech(d*x+c)/b^3/(a^2+b^2)/d^2-1/2*a^3*(f*x+e)*sech
(d*x+c)^2/b^2/(a^2+b^2)/d+1/2*a^3*f*tanh(d*x+c)/b^2/(a^2+b^2)/d^2+1/2*a^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b^3/
d-1/2*I*a^2*f*polylog(2,-I*exp(d*x+c))/b^3/d^2-a^4*(f*x+e)*arctan(exp(d*x+c))/b^3/(a^2+b^2)/d+1/2*a^3*f*polylo
g(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2+1/2*a^2*f*sech(d*x+c)/b^3/d^2+1/2*a*(f*x+e)*sech(d*x+c)^2/b^2/d-1/2*a*f*t
anh(d*x+c)/b^2/d^2+(f*x+e)*arctan(exp(d*x+c))/b/d+1/2*I*a^4*f*polylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+a^2*(
f*x+e)*arctan(exp(d*x+c))/b^3/d+a^3*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(
a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-a^3*f*polyl
og(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^
2+b^2)^2/d^2-1/2*f*sech(d*x+c)/b/d^2-1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b/d-1/2*I*f*polylog(2,-I*exp(d*x+c))/
b/d^2+1/2*I*f*polylog(2,I*exp(d*x+c))/b/d^2+I*a^4*f*polylog(2,-I*exp(d*x+c))/b/(a^2+b^2)^2/d^2+1/2*I*a^2*f*pol
ylog(2,I*exp(d*x+c))/b^3/d^2-1/2*a^4*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b^3/(a^2+b^2)/d-I*a^4*f*polylog(2,I*exp(d
*x+c))/b/(a^2+b^2)^2/d^2-1/2*I*a^4*f*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2

________________________________________________________________________________________

Rubi [A]
time = 1.11, antiderivative size = 894, normalized size of antiderivative = 1.00, 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 = {5686, 5563, 4265, 2317, 2438, 4270, 5702, 5559, 3852, 8, 5692, 5680, 2221, 6874, 3799} \begin {gather*} -\frac {(e+f x) \text {ArcTan}\left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}-\frac {2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d}+\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d^2}-\frac {i f \text {Li}_2\left (i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {i f \text {Li}_2\left (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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_2\left (-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) \text {ArcTan}\left (e^{c+d x}\right ) a^2}{b^3 d}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{2 b^3 d^2}+\frac {i f \text {Li}_2\left (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) \text {ArcTan}\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 {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} \end {gather*}

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 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 3799

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 3852

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 4265

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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(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 5563

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 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 5686

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*Sech
[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 5692

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 5702

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 6874

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

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) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 b^2 d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^2}+\frac {(i f) \text {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) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b d^2}-\frac {(i f) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 5.47, size = 588, normalized size = 0.66 \begin {gather*} \frac {-2 a^3 d e (c+d x)+2 a^3 c f (c+d x)+6 a^2 b d e \text {ArcTan}\left (e^{c+d x}\right )+2 b^3 d e \text {ArcTan}\left (e^{c+d x}\right )-6 a^2 b c f \text {ArcTan}\left (e^{c+d x}\right )-2 b^3 c f \text {ArcTan}\left (e^{c+d x}\right )+3 i a^2 b 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 )-3 i a^2 b 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 a^3 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 a^3 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-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 \left (3 a^2+b^2\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (3 a^2+b^2\right ) f \text {PolyLog}\left (2,i e^{c+d x}\right )-2 a^3 f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 a^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+a^3 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-\left (a^2+b^2\right ) f \text {sech}(c+d x) (b+a \sinh (c+d x))+\left (a^2+b^2\right ) d (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (820 ) = 1640\).
time = 5.72, size = 2284, normalized size = 2.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(2284\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

-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)) - (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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5468 vs. \(2 (805) = 1610\).
time = 0.53, size = 5468, normalized size = 6.12 \begin {gather*} \text {Too large to display} \end {gather*}

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*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (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*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (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*cosh(1) + 2*(a^3 + a*b^2)*d*sinh(1) + (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*cosh(1) + 2*(a^3 + a*b^2)*d*sinh(1) + (a^3 + a*
b^2)*f - 3*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (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*cosh(1) + (a^2*b + b^3)
*d*sinh(1) - (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*co
sh(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 + 4*(2*a^3*f + 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 + 2*(2*a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 2
*(2*a^3*f + 3*(2*a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + I*(3*a^2*b + b^3)*f)*sinh(d*x + c)^2 + I*(3*a^
2*b + b^3)*f + 4*((2*a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^3 + (2*a^3*f + 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 + 4*(2*a^3*f - 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 + 2*(2*a^3*f - I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 2*(2*a^3*f + 3*(2*a^3*f - I*(3*a^2*b +
b^3)*f)*cosh(d*x + c)^2 - I*(3*a^2*b + b^3)*f)*sinh(d*x + c)^2 - I*(3*a^2*b + b^3)*f + 4*((2*a^3*f - I*(3*a^2
*b + b^3)*f)*cosh(d*x + c)^3 + (2*a^3*f - 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*c*f - a^3*d*cosh(1) + (a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c
)^4 - a^3*d*sinh(1) + 4*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*c*f - a
^3*d*cosh(1) - a^3*d*sinh(1))*sinh(d*x + c)^4 + 2*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^2 +
2*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1) + 3*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^2)*sinh
(d*x + c)^2 + 4*((a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^3 + (a^3*c*f - a^3*d*cosh(1) - a^3*d*
sinh(1))*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*c*f - a^3*d*cosh(1) + (a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^4 - a^3*d*sinh(1)
+ 4*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*c*f - a^3*d*cosh(1) - a^3*
d*sinh(1))*sinh(d*x + c)^4 + 2*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^2 + 2*(a^3*c*f - a^3*d*
cosh(1) - a^3*d*sinh(1) + 3*(a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a
^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c)^3 + (a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*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)*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*c*f - 2*a^
3*d*cosh(1) + (2*a^3*c*f - 2*a^3*d*cosh(1) - 2*a^3*d*sinh(1) + I*(3*a^2*b + b^3)*c*f - I*(3*a^2*b + b^3)*d*cos
h(1) - I*(3*a^2*b + b^3)*d*sinh(1))*cosh(d*x + c)^4 - 2*a^3*d*sinh(1) + 4*(2*a^3*c*f - 2*a^3*d*cosh(1) - 2*a^3
*d*sinh(1) + I*(3*a^2*b + b^3)*c*f - I*(3*a^2*b...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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