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

Optimal. Leaf size=711 \[ \frac {i a^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {i a^2 b f \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i a^2 f \text {Li}_2\left (i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}-\frac {i a^2 b f \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {a b^2 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}+\frac {a f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac {a^2 f \text {sech}(c+d x)}{2 b d^2 \left (a^2+b^2\right )}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {a b^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d \left (a^2+b^2\right )}-\frac {2 a^2 b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}-\frac {a (e+f x) \text {sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d \left (a^2+b^2\right )}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b d^2}+\frac {i f \text {Li}_2\left (i e^{c+d x}\right )}{2 b d^2}+\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d} \]

[Out]

(f*x+e)*arctan(exp(d*x+c))/b/d-2*a^2*b*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d-a^2*(f*x+e)*arctan(exp(d*x+c))
/b/(a^2+b^2)/d+a*b^2*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d-a*b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(
1/2)))/(a^2+b^2)^2/d-a*b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+I*a^2*b*f*polylog(2,-I
*exp(d*x+c))/(a^2+b^2)^2/d^2-1/2*I*a^2*f*polylog(2,I*exp(d*x+c))/b/(a^2+b^2)/d^2+1/2*I*a^2*f*polylog(2,-I*exp(
d*x+c))/b/(a^2+b^2)/d^2+1/2*I*f*polylog(2,I*exp(d*x+c))/b/d^2-I*a^2*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^
2-1/2*I*f*polylog(2,-I*exp(d*x+c))/b/d^2+1/2*a*b^2*f*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2-a*b^2*f*polylo
g(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a*b^2*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*a^2*f*sech(d*x+c)/b/(a^2+b^2)/d^2-1/2*a*(f*x+e)*sech(d*x+c)^2/(a^2+b
^2)/d+1/2*a*f*tanh(d*x+c)/(a^2+b^2)/d^2+1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b/d-1/2*a^2*(f*x+e)*sech(d*x+c)*ta
nh(d*x+c)/b/(a^2+b^2)/d

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Rubi [A]  time = 0.99, antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5583, 4185, 4180, 2279, 2391, 5573, 5561, 2190, 6742, 3718, 5451, 3767, 8} \[ \frac {i a^2 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {i a^2 b f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i a^2 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}-\frac {i a^2 b f \text {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {a b^2 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}-\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}+\frac {a f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac {a^2 f \text {sech}(c+d x)}{2 b d^2 \left (a^2+b^2\right )}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {a b^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d \left (a^2+b^2\right )}-\frac {2 a^2 b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}-\frac {a (e+f x) \text {sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d \left (a^2+b^2\right )}+\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 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 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 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 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 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 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 6742

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

Rubi steps

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

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Mathematica [A]  time = 7.98, size = 587, normalized size = 0.83 \[ \frac {b \left (i f \left (a^2-b^2\right ) \text {Li}_2\left (-i e^{c+d x}\right )-i f \left (a^2-b^2\right ) \text {Li}_2\left (i e^{c+d x}\right )-2 a^2 d e \tan ^{-1}\left (e^{c+d x}\right )-i a^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+i a^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^2 c f \tan ^{-1}\left (e^{c+d x}\right )-2 a b d e (c+d x)+2 a b d e \log \left (e^{2 (c+d x)}+1\right )+a b f \text {Li}_2\left (-e^{2 (c+d x)}\right )-a b f (c+d x)^2+2 a b c f (c+d x)-2 a b c f \log \left (e^{2 (c+d x)}+1\right )+2 a b f (c+d x) \log \left (e^{2 (c+d x)}+1\right )+2 b^2 d e \tan ^{-1}\left (e^{c+d x}\right )+i b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )-i b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )-2 b^2 c f \tan ^{-1}\left (e^{c+d x}\right )\right )-2 a b^2 \left (f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac {1}{2} f (c+d x)^2\right )+d \left (a^2+b^2\right ) (e+f x) \text {sech}^2(c+d x) (b \sinh (c+d x)-a)+f \left (a^2+b^2\right ) \text {sech}(c+d x) (a \sinh (c+d x)+b)}{2 d^2 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b^2*(-1/2*(f*(c + d*x)^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*Sinh[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 + Sqrt[a^2 + b^2]))]) +
b*(-2*a*b*d*e*(c + d*x) + 2*a*b*c*f*(c + d*x) - a*b*f*(c + d*x)^2 - 2*a^2*d*e*ArcTan[E^(c + d*x)] + 2*b^2*d*e
*ArcTan[E^(c + d*x)] + 2*a^2*c*f*ArcTan[E^(c + d*x)] - 2*b^2*c*f*ArcTan[E^(c + d*x)] - I*a^2*f*(c + d*x)*Log[1
- I*E^(c + d*x)] + I*b^2*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + I*a^2*f*(c + d*x)*Log[1 + I*E^(c + d*x)] - I*b^
2*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a*b*d*e*Log[1 + E^(2*(c + d*x))] - 2*a*b*c*f*Log[1 + E^(2*(c + d*x))]
+ 2*a*b*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + I*(a^2 - b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] - I*(a^2 - b^2)*f
*PolyLog[2, I*E^(c + d*x)] + a*b*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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fricas [B]  time = 0.71, size = 4963, normalized size = 6.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sech(d*x+c)^2*tanh(d*x+c)/(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*b^2*f*cosh(d*x + c)^4 + 4*a*b^2*f*co
sh(d*x + c)*sinh(d*x + c)^3 + a*b^2*f*sinh(d*x + c)^4 + 2*a*b^2*f*cosh(d*x + c)^2 + a*b^2*f + 2*(3*a*b^2*f*cos
h(d*x + c)^2 + a*b^2*f)*sinh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c)^3 + a*b^2*f*cosh(d*x + c))*sinh(d*x + c))*d
ilog((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*cosh(d*x + c)^4 + 4*a*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*f*sinh(d*x + c)^4 + 2*a*b^2*f
*cosh(d*x + c)^2 + a*b^2*f + 2*(3*a*b^2*f*cosh(d*x + c)^2 + a*b^2*f)*sinh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c
)^3 + a*b^2*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*si
nh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + ((2*a*b^2*f - I*(a^2*b - b^3)*f)*cosh(d*x + c)^4 + (8*a*b^2*f
- 4*I*(a^2*b - b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*f - I*(a^2*b - b^3)*f)*sinh(d*x + c)^4 + 2*a*
b^2*f + (4*a*b^2*f - 2*I*(a^2*b - b^3)*f)*cosh(d*x + c)^2 + (4*a*b^2*f + (12*a*b^2*f - 6*I*(a^2*b - b^3)*f)*co
sh(d*x + c)^2 - 2*I*(a^2*b - b^3)*f)*sinh(d*x + c)^2 - I*(a^2*b - b^3)*f + ((8*a*b^2*f - 4*I*(a^2*b - b^3)*f)*
cosh(d*x + c)^3 + (8*a*b^2*f - 4*I*(a^2*b - b^3)*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*si
nh(d*x + c)) + ((2*a*b^2*f + I*(a^2*b - b^3)*f)*cosh(d*x + c)^4 + (8*a*b^2*f + 4*I*(a^2*b - b^3)*f)*cosh(d*x +
c)*sinh(d*x + c)^3 + (2*a*b^2*f + I*(a^2*b - b^3)*f)*sinh(d*x + c)^4 + 2*a*b^2*f + (4*a*b^2*f + 2*I*(a^2*b -
b^3)*f)*cosh(d*x + c)^2 + (4*a*b^2*f + (12*a*b^2*f + 6*I*(a^2*b - b^3)*f)*cosh(d*x + c)^2 + 2*I*(a^2*b - b^3)*
f)*sinh(d*x + c)^2 + I*(a^2*b - b^3)*f + ((8*a*b^2*f + 4*I*(a^2*b - b^3)*f)*cosh(d*x + c)^3 + (8*a*b^2*f + 4*I
*(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*b^2*d*e - a*b
^2*c*f + (a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (
a*b^2*d*e - a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*e - a*b^2*c*f
+ 3*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^3 + (a
*b^2*d*e - a*b^2*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*b^2*d*e - a*b^2*c*f + (a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*e - a*b^2
*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*e - a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*e - a*b^2*c*f)*cosh
(d*x + c)^2 + 2*(a*b^2*d*e - a*b^2*c*f + 3*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^
2*d*e - a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*e - a*b^2*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*b^2*d*f*x + a*b^2*c*f + (a*b^2*d*f*x + a*b^2*
c*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*c*f)*s
inh(d*x + c)^4 + 2*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f*x + a*b^2*c*f + 3*(a*b^2*d*f*x + a
*b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*f*x + a*b
^2*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*b^2*d*f*x + a*b^2*c*f + (a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^4 +
4*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*
b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f*x + a*b^2*c*f + 3*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*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*b^2*d*e - 2*a*b^2*c*f + (2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3
)*c*f)*cosh(d*x + c)^4 + (8*a*b^2*d*e - 8*a*b^2*c*f - 4*I*(a^2*b - b^3)*d*e + 4*I*(a^2*b - b^3)*c*f)*cosh(d*x
+ c)*sinh(d*x + c)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*sinh(d*x + c)^4
- I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f + (4*a*b^2*d*e - 4*a*b^2*c*f - 2*I*(a^2*b - b^3)*d*e + 2*I*(a^2*b
- b^3)*c*f)*cosh(d*x + c)^2 + (4*a*b^2*d*e - 4*a*b^2*c*f - 2*I*(a^2*b - b^3)*d*e + 2*I*(a^2*b - b^3)*c*f + (1
2*a*b^2*d*e - 12*a*b^2*c*f - 6*I*(a^2*b - b^3)*d*e + 6*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
((8*a*b^2*d*e - 8*a*b^2*c*f - 4*I*(a^2*b - b^3)*d*e + 4*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (8*a*b^2*d*e -
8*a*b^2*c*f - 4*I*(a^2*b - b^3)*d*e + 4*I*(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*b^2*d*e - 2*a*b^2*c*f + (2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*
b - b^3)*c*f)*cosh(d*x + c)^4 + (8*a*b^2*d*e - 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*e - 4*I*(a^2*b - b^3)*c*f)*co
sh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*sinh(d*x
+ c)^4 + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f + (4*a*b^2*d*e - 4*a*b^2*c*f + 2*I*(a^2*b - b^3)*d*e - 2*I
*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2 + (4*a*b^2*d*e - 4*a*b^2*c*f + 2*I*(a^2*b - b^3)*d*e - 2*I*(a^2*b - b^3)*c
*f + (12*a*b^2*d*e - 12*a*b^2*c*f + 6*I*(a^2*b - b^3)*d*e - 6*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + ((8*a*b^2*d*e - 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*e - 4*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (8*a*b^
2*d*e - 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*e - 4*I*(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*b^2*d*f*x + 2*a*b^2*c*f + (2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*
f*x + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^4 + (8*a*b^2*d*f*x + 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*f*x + 4*I*(a^2
*b - b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b
- b^3)*c*f)*sinh(d*x + c)^4 + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f + (4*a*b^2*d*f*x + 4*a*b^2*c*f + 2*
I*(a^2*b - b^3)*d*f*x + 2*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2 + (4*a*b^2*d*f*x + 4*a*b^2*c*f + 2*I*(a^2*b - b
^3)*d*f*x + 2*I*(a^2*b - b^3)*c*f + (12*a*b^2*d*f*x + 12*a*b^2*c*f + 6*I*(a^2*b - b^3)*d*f*x + 6*I*(a^2*b - b^
3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d*f*x + 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*f*x + 4*I*(a^2*
b - b^3)*c*f)*cosh(d*x + c)^3 + (8*a*b^2*d*f*x + 8*a*b^2*c*f + 4*I*(a^2*b - b^3)*d*f*x + 4*I*(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*b^2*d*f*x + 2*a*b^2*c*f + (2
*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^4 + (8*a*b^2*d*f*x + 8
*a*b^2*c*f - 4*I*(a^2*b - b^3)*d*f*x - 4*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d*f*x +
2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c*f)*sinh(d*x + c)^4 - I*(a^2*b - b^3)*d*f*x - I*(a^2*b
- b^3)*c*f + (4*a*b^2*d*f*x + 4*a*b^2*c*f - 2*I*(a^2*b - b^3)*d*f*x - 2*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2
+ (4*a*b^2*d*f*x + 4*a*b^2*c*f - 2*I*(a^2*b - b^3)*d*f*x - 2*I*(a^2*b - b^3)*c*f + (12*a*b^2*d*f*x + 12*a*b^2*
c*f - 6*I*(a^2*b - b^3)*d*f*x - 6*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d*f*x + 8*
a*b^2*c*f - 4*I*(a^2*b - b^3)*d*f*x - 4*I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (8*a*b^2*d*f*x + 8*a*b^2*c*f -
4*I*(a^2*b - b^3)*d*f*x - 4*I*(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*c
osh(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*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^
2*cosh(d*x + c))*sinh(d*x + c))

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.28, size = 2074, normalized size = 2.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a^2+b^2)^(3/2)/d*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/(a^2+b^2)^(1/2)/d
*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))+I/d*b/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(
1+I*exp(d*x+c))*x+I/d^2*b/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c+1/(a^2+b^2)^(3/2)/d^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*b/(a^2+b^2)*a^2*e/(2*a^2+2*b^2)*arctan(ex
p(d*x+c))-I/d^2*b^3/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+2/d*b^3/(a^2+b^2)*e/(2*a^2+2*b^2)*arctan(e
xp(d*x+c))-I/d*b/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-I/d^2*b/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1
-I*exp(d*x+c))*c-(-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+1/(a^2+b^2)^(3/2)/d^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))-1/(a^2+b^2)^(3/2)/d*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/(a^2+b^2)^
(1/2)/d^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*b^2/(a^2+b^2)*f/(2*a^2+2
*b^2)*ln(1-I*exp(d*x+c))*a*x+2/d^2*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*c+I/d^2*b^3/(a^2+b^2)*f/
(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c+2/d*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*a*x-2/d*b^2/(a^2+b^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*x-2/d^2*b^2/(a^2+b^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*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln((b*ex
p(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a*x-2/d^2*b^2/(a^2+b^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^2*b^2/(a^2+b^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*b^2/(a^2+b^2)*f*c/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))-I/d*b^3/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*
exp(d*x+c))*x-I/d^2*b^3/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-I/d^2*b/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*d
ilog(1-I*exp(d*x+c))+2/d^2*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*a*c+I/d^2*b/(a^2+b^2)*a^2*f/(2*a^2
+2*b^2)*dilog(1+I*exp(d*x+c))+2/d^2*b/(a^2+b^2)*a^2*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))+I/d*b^3/(a^2+b^2)*f/(
2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-2/d^2*b^3/(a^2+b^2)*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2/d^2*b^2/(a^2+b^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-2/d^2*b^2/(a^2+b^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+2/d^2*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog
(1+I*exp(d*x+c))*a+2/d^2*b^2/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))*a+I/d^2*b^3/(a^2+b^2)*f/(2*a^2+2*
b^2)*dilog(1-I*exp(d*x+c))-2/d*b^2/(a^2+b^2)*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*b^2/(
a^2+b^2)*e/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (\frac {a b^{2} \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 b^{2} \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 (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 )}} + 4 \, \int -\frac {a^{2} b^{2} x e^{\left (d x + c\right )} - a b^{3} x}{2 \, {\left (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 )}\right )}}\,{d x} - 4 \, \int \frac {2 \, a b^{2} x + {\left (a^{2} b e^{c} - b^{3} e^{c}\right )} x e^{\left (d x\right )}}{4 \, {\left (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 )}\right )}}\,{d x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^2*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*b^2*log(e^(-2*d*x - 2*
c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (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)) + 4*integrate(-1/2*(a^2*b^2*x*e^(d*x + c) - a*b^3*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) - 4*integrate(1/4*(2*a*b^2*x + (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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right ) \tanh {\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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