\(\int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [213]

Optimal result

Integrand size = 29, antiderivative size = 163 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

2*I*(f*x+e)*arctanh(exp(d*x+c))/a/d-(f*x+e)*coth(d*x+c)/a/d+2*f*ln(cosh(1/2*c+1/4*I*Pi+1/2*d*x))/a/d^2+f*ln(si
nh(d*x+c))/a/d^2+I*f*polylog(2,-exp(d*x+c))/a/d^2-I*f*polylog(2,exp(d*x+c))/a/d^2-(f*x+e)*tanh(1/2*c+1/4*I*Pi+
1/2*d*x)/a/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5694, 4269, 3556, 4267, 2317, 2438, 3399} \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d} \]

[In]

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

[Out]

((2*I)*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (2*f*Log[Cosh[c/2 + (I/4)*Pi
+ (d*x)/2]])/(a*d^2) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (I*f*PolyLog[2, -E^(c + d*x)])/(a*d^2) - (I*f*PolyLog[
2, E^(c + d*x)])/(a*d^2) - ((e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3556

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

Rule 4267

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

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5694

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

Rubi steps \begin{align*} \text {integral}& = -\left (i \int \frac {(e+f x) \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x) \text {csch}^2(c+d x) \, dx}{a} \\ & = -\frac {(e+f x) \coth (c+d x)}{a d}-\frac {i \int (e+f x) \text {csch}(c+d x) \, dx}{a}+\frac {f \int \coth (c+d x) \, dx}{a d}-\int \frac {e+f x}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {\int (e+f x) \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}+\frac {(i f) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(i f) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d} \\ & = \frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}+\frac {f \int \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(163)=326\).

Time = 3.08 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.25 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (-d (e+f x) \cosh \left (\frac {1}{2} (c+d x)\right ) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right )+4 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 f \log (\cosh (c+d x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (f (c+d x)+(f-i d (e+f x)) \log \left (1-e^{-c-d x}\right )+(f+i d (e+f x)) \log \left (1+e^{-c-d x}\right )-i f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+i f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )-4 d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right )+2 f (c+d x) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )-i d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right ) \left (-i+\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 d^2 (a+i a \sinh (c+d x))} \]

[In]

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

[Out]

((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(-(d*(e + f*x)*Cosh[(c + d*x)/2]*(I + Coth[(c + d*x)/2])) + (4*I)*f
*ArcTan[Tanh[(c + d*x)/2]]*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + 2*f*Log[Cosh[c + d*x]]*(Cosh[(c + d*x)/
2] + I*Sinh[(c + d*x)/2]) + 2*(f*(c + d*x) + (f - I*d*(e + f*x))*Log[1 - E^(-c - d*x)] + (f + I*d*(e + f*x))*L
og[1 + E^(-c - d*x)] - I*f*PolyLog[2, -E^(-c - d*x)] + I*f*PolyLog[2, E^(-c - d*x)])*(Cosh[(c + d*x)/2] + I*Si
nh[(c + d*x)/2]) - 4*d*(e + f*x)*Sinh[(c + d*x)/2] + 2*f*(c + d*x)*((-I)*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]
) - I*d*(e + f*x)*Sinh[(c + d*x)/2]*(-I + Tanh[(c + d*x)/2])))/(2*d^2*(a + I*a*Sinh[c + d*x]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (143 ) = 286\).

Time = 2.07 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.05

method	result	size
risch	\(-\frac {2 i \left (f x \,{\mathrm e}^{2 d x +2 c}+e \,{\mathrm e}^{2 d x +2 c}-2 f x -i {\mathrm e}^{d x +c} f x -2 e -i {\mathrm e}^{d x +c} e \right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{d x +c}-i\right ) a d}+\frac {i c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}-\frac {i e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}+\frac {i e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}-\frac {i f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{d^{2} a}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}+\frac {f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{2} a}-\frac {4 f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {i f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{d a}+\frac {i f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}+\frac {2 i f \arctan \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {i f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {i f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}\)	\(334\)

[In]

int((f*x+e)*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2*I*(f*x*exp(2*d*x+2*c)+e*exp(2*d*x+2*c)-2*f*x-I*exp(d*x+c)*f*x-2*e-I*exp(d*x+c)*e)/(exp(2*d*x+2*c)-1)/(exp(d
*x+c)-I)/a/d+I/d^2/a*c*f*ln(exp(d*x+c)-1)-I/d/a*e*ln(exp(d*x+c)-1)+I/d/a*e*ln(exp(d*x+c)+1)-I/d^2/a*f*ln(1-exp
(d*x+c))*c+1/d^2/a*f*ln(exp(d*x+c)-1)+1/d^2/a*f*ln(exp(d*x+c)+1)+1/d^2/a*f*ln(1+exp(2*d*x+2*c))-4/d^2/a*f*ln(e
xp(d*x+c))-I/d/a*f*ln(1-exp(d*x+c))*x+I/d/a*f*ln(exp(d*x+c)+1)*x+2*I/d^2/a*f*arctan(exp(d*x+c))-I*f*polylog(2,
exp(d*x+c))/a/d^2+I*f*polylog(2,-exp(d*x+c))/a/d^2

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (139) = 278\).

Time = 0.27 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.10 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {4 i \, d e - 2 i \, c f + {\left (i \, f e^{\left (3 \, d x + 3 \, c\right )} + f e^{\left (2 \, d x + 2 \, c\right )} - i \, f e^{\left (d x + c\right )} - f\right )} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) + {\left (-i \, f e^{\left (3 \, d x + 3 \, c\right )} - f e^{\left (2 \, d x + 2 \, c\right )} + i \, f e^{\left (d x + c\right )} + f\right )} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\left (2 \, d f x + c f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 2 \, {\left (-i \, d f x + i \, d e - i \, c f\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d f x - d e + c f\right )} e^{\left (d x + c\right )} - {\left (d f x + d e - {\left (i \, d f x + i \, d e + f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f x + d e - i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d f x - i \, d e - f\right )} e^{\left (d x + c\right )} - i \, f\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, {\left (f e^{\left (3 \, d x + 3 \, c\right )} - i \, f e^{\left (2 \, d x + 2 \, c\right )} - f e^{\left (d x + c\right )} + i \, f\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + {\left (d e - {\left (c - i\right )} f + {\left (-i \, d e + {\left (i \, c + 1\right )} f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d e - {\left (c - i\right )} f\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, d e + {\left (-i \, c - 1\right )} f\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) + {\left (d f x + c f + {\left (-i \, d f x - i \, c f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f x + c f\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, d f x + i \, c f\right )} e^{\left (d x + c\right )}\right )} \log \left (-e^{\left (d x + c\right )} + 1\right )}{a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )} - a d^{2} e^{\left (d x + c\right )} + i \, a d^{2}} \]

[In]

integrate((f*x+e)*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(4*I*d*e - 2*I*c*f + (I*f*e^(3*d*x + 3*c) + f*e^(2*d*x + 2*c) - I*f*e^(d*x + c) - f)*dilog(-e^(d*x + c)) + (-I
*f*e^(3*d*x + 3*c) - f*e^(2*d*x + 2*c) + I*f*e^(d*x + c) + f)*dilog(e^(d*x + c)) - 2*(2*d*f*x + c*f)*e^(3*d*x
+ 3*c) - 2*(-I*d*f*x + I*d*e - I*c*f)*e^(2*d*x + 2*c) + 2*(d*f*x - d*e + c*f)*e^(d*x + c) - (d*f*x + d*e - (I*
d*f*x + I*d*e + f)*e^(3*d*x + 3*c) - (d*f*x + d*e - I*f)*e^(2*d*x + 2*c) - (-I*d*f*x - I*d*e - f)*e^(d*x + c)
- I*f)*log(e^(d*x + c) + 1) + 2*(f*e^(3*d*x + 3*c) - I*f*e^(2*d*x + 2*c) - f*e^(d*x + c) + I*f)*log(e^(d*x + c
) - I) + (d*e - (c - I)*f + (-I*d*e + (I*c + 1)*f)*e^(3*d*x + 3*c) - (d*e - (c - I)*f)*e^(2*d*x + 2*c) + (I*d*
e + (-I*c - 1)*f)*e^(d*x + c))*log(e^(d*x + c) - 1) + (d*f*x + c*f + (-I*d*f*x - I*c*f)*e^(3*d*x + 3*c) - (d*f
*x + c*f)*e^(2*d*x + 2*c) + (I*d*f*x + I*c*f)*e^(d*x + c))*log(-e^(d*x + c) + 1))/(a*d^2*e^(3*d*x + 3*c) - I*a
*d^2*e^(2*d*x + 2*c) - a*d^2*e^(d*x + c) + I*a*d^2)

Sympy [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]

[In]

integrate((f*x+e)*csch(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*(Integral(e*csch(c + d*x)**2/(sinh(c + d*x) - I), x) + Integral(f*x*csch(c + d*x)**2/(sinh(c + d*x) - I), x
))/a

Maxima [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((f*x + e)*csch(d*x + c)^2/(I*a*sinh(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)), x)