\(\int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [219]

Optimal result

Integrand size = 29, antiderivative size = 214 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {i f \log (\sinh (c+d x))}{a d^2}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

3*(f*x+e)*arctanh(exp(d*x+c))/a/d+I*(f*x+e)*coth(d*x+c)/a/d-1/2*f*csch(d*x+c)/a/d^2-1/2*(f*x+e)*coth(d*x+c)*cs
ch(d*x+c)/a/d-2*I*f*ln(cosh(1/2*c+1/4*I*Pi+1/2*d*x))/a/d^2-I*f*ln(sinh(d*x+c))/a/d^2+3/2*f*polylog(2,-exp(d*x+
c))/a/d^2-3/2*f*polylog(2,exp(d*x+c))/a/d^2+I*(f*x+e)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5694, 4270, 4267, 2317, 2438, 4269, 3556, 3399} \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {i f \log (\sinh (c+d x))}{a d^2}-\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[In]

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

[Out]

(3*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) + (I*(e + f*x)*Coth[c + d*x])/(a*d) - (f*Csch[c + d*x])/(2*a*d^2) - (
(e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - ((2*I)*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) - (I*f*
Log[Sinh[c + d*x]])/(a*d^2) + (3*f*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*f*PolyLog[2, E^(c + d*x)])/(2*a*d^
2) + (I*(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 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 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}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x) \text {csch}^3(c+d x) \, dx}{a} \\ & = -\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {i \int (e+f x) \text {csch}^2(c+d x) \, dx}{a}-\frac {\int (e+f x) \text {csch}(c+d x) \, dx}{2 a}-\int \frac {(e+f x) \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {(e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}+i \int \frac {e+f x}{a+i a \sinh (c+d x)} \, dx-\frac {\int (e+f x) \text {csch}(c+d x) \, dx}{a}-\frac {(i f) \int \coth (c+d x) \, dx}{a d}+\frac {f \int \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac {f \int \log \left (1+e^{c+d x}\right ) \, dx}{2 a d} \\ & = \frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {i f \log (\sinh (c+d x))}{a d^2}+\frac {i \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 {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}+\frac {f \int \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {f \int \log \left (1+e^{c+d x}\right ) \, dx}{a d} \\ & = \frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {i f \log (\sinh (c+d x))}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}-\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}-\frac {(i f) \int \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {i f \log (\sinh (c+d x))}{a d^2}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {i (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. \(461\) vs. \(2(214)=428\).

Time = 3.54 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (2 i (i f+2 d (e+f x)) \cosh \left (\frac {1}{2} (c+d x)\right ) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right )-d (e+f x) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )-8 f (c+d x) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+16 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+4 \left (-2 i f (c+d x)-(2 i f+3 d (e+f x)) \log \left (1-e^{-c-d x}\right )+(-2 i f+3 d (e+f x)) \log \left (1+e^{-c-d x}\right )-3 f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+3 f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+16 i d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right )+8 f \log (\cosh (c+d x)) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 (f+2 i d (e+f x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \tanh \left (\frac {1}{2} (c+d x)\right )-i d (e+f x) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-i+\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 d^2 (a+i a \sinh (c+d x))} \]

[In]

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

[Out]

((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*((2*I)*(I*f + 2*d*(e + f*x))*Cosh[(c + d*x)/2]*(I + Coth[(c + d*x)/
2]) - d*(e + f*x)*(I + Coth[(c + d*x)/2])*Csch[(c + d*x)/2] - 8*f*(c + d*x)*(Cosh[(c + d*x)/2] + I*Sinh[(c + d
*x)/2]) + 16*f*ArcTan[Tanh[(c + d*x)/2]]*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + 4*((-2*I)*f*(c + d*x) - (
(2*I)*f + 3*d*(e + f*x))*Log[1 - E^(-c - d*x)] + ((-2*I)*f + 3*d*(e + f*x))*Log[1 + E^(-c - d*x)] - 3*f*PolyLo
g[2, -E^(-c - d*x)] + 3*f*PolyLog[2, E^(-c - d*x)])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + (16*I)*d*(e +
f*x)*Sinh[(c + d*x)/2] + 8*f*Log[Cosh[c + d*x]]*((-I)*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]) + 2*(f + (2*I)*d*
(e + f*x))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*Tanh[(c + d*x)/2] - I*d*(e + f*x)*Sech[(c + d*x)/2]*(-I +
Tanh[(c + d*x)/2])))/(8*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. 440 vs. \(2 (185 ) = 370\).

Time = 2.38 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.06

method	result	size
risch	\(-\frac {-3 i d e \,{\mathrm e}^{3 d x +3 c}-5 \,{\mathrm e}^{2 d x +2 c} d f x +3 d f x \,{\mathrm e}^{4 d x +4 c}+i d e \,{\mathrm e}^{d x +c}-5 \,{\mathrm e}^{2 d x +2 c} d e +3 d e \,{\mathrm e}^{4 d x +4 c}+i d f x \,{\mathrm e}^{d x +c}+4 d f x +f \,{\mathrm e}^{4 d x +4 c}-i {\mathrm e}^{3 d x +3 c} f +i {\mathrm e}^{d x +c} f +4 d e -f \,{\mathrm e}^{2 d x +2 c}-3 i d f x \,{\mathrm e}^{3 d x +3 c}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d^{2} \left ({\mathrm e}^{d x +c}-i\right ) a}-\frac {3 f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{2 a d}+\frac {3 f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{2 a d}-\frac {3 e \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {4 i f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {i f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {i f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {i f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}+\frac {3 e \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}-\frac {3 f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{2 a \,d^{2}}+\frac {3 c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{2}}+\frac {2 f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {3 f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}-\frac {3 f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}\)	\(441\)

[In]

int((f*x+e)*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-(-3*I*d*e*exp(3*d*x+3*c)-5*exp(2*d*x+2*c)*d*f*x+3*d*f*x*exp(4*d*x+4*c)+I*d*e*exp(d*x+c)-5*exp(2*d*x+2*c)*d*e+
3*d*e*exp(4*d*x+4*c)+I*d*f*x*exp(d*x+c)+4*d*f*x+f*exp(4*d*x+4*c)-I*exp(3*d*x+3*c)*f+I*exp(d*x+c)*f+4*d*e-f*exp
(2*d*x+2*c)-3*I*d*f*x*exp(3*d*x+3*c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/a-3/2/a/d*f*ln(1-exp(d*x+c))*x+3
/2/a/d*f*ln(exp(d*x+c)+1)*x-3/2/a/d*e*ln(exp(d*x+c)-1)+4*I/a/d^2*f*ln(exp(d*x+c))-I/a/d^2*f*ln(exp(d*x+c)-1)-I
/a/d^2*f*ln(exp(d*x+c)+1)-I/a/d^2*f*ln(1+exp(2*d*x+2*c))+3/2/a/d*e*ln(exp(d*x+c)+1)-3/2/a/d^2*f*ln(1-exp(d*x+c
))*c+3/2/a/d^2*c*f*ln(exp(d*x+c)-1)+2/a/d^2*f*arctan(exp(d*x+c))+3/2*f*polylog(2,-exp(d*x+c))/a/d^2-3/2*f*poly
log(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. 817 vs. \(2 (181) = 362\).

Time = 0.28 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.82 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(8*d*e - 4*c*f - 3*(f*e^(5*d*x + 5*c) - I*f*e^(4*d*x + 4*c) - 2*f*e^(3*d*x + 3*c) + 2*I*f*e^(2*d*x + 2*c)
+ f*e^(d*x + c) - I*f)*dilog(-e^(d*x + c)) + 3*(f*e^(5*d*x + 5*c) - I*f*e^(4*d*x + 4*c) - 2*f*e^(3*d*x + 3*c)
+ 2*I*f*e^(2*d*x + 2*c) + f*e^(d*x + c) - I*f)*dilog(e^(d*x + c)) + 4*(-2*I*d*f*x - I*c*f)*e^(5*d*x + 5*c) -
2*(d*f*x - 3*d*e + (2*c - 1)*f)*e^(4*d*x + 4*c) + 2*(5*I*d*f*x - 3*I*d*e + (4*I*c - I)*f)*e^(3*d*x + 3*c) + 2*
(3*d*f*x - 5*d*e + (4*c - 1)*f)*e^(2*d*x + 2*c) + 2*(-3*I*d*f*x + I*d*e + (-2*I*c + I)*f)*e^(d*x + c) - (-3*I*
d*f*x - 3*I*d*e + (3*d*f*x + 3*d*e - 2*I*f)*e^(5*d*x + 5*c) + (-3*I*d*f*x - 3*I*d*e - 2*f)*e^(4*d*x + 4*c) - 2
*(3*d*f*x + 3*d*e - 2*I*f)*e^(3*d*x + 3*c) - 2*(-3*I*d*f*x - 3*I*d*e - 2*f)*e^(2*d*x + 2*c) + (3*d*f*x + 3*d*e
- 2*I*f)*e^(d*x + c) - 2*f)*log(e^(d*x + c) + 1) + 4*(I*f*e^(5*d*x + 5*c) + f*e^(4*d*x + 4*c) - 2*I*f*e^(3*d*
x + 3*c) - 2*f*e^(2*d*x + 2*c) + I*f*e^(d*x + c) + f)*log(e^(d*x + c) - I) - (3*I*d*e + (-3*I*c - 2)*f - (3*d*
e - (3*c - 2*I)*f)*e^(5*d*x + 5*c) + (3*I*d*e + (-3*I*c - 2)*f)*e^(4*d*x + 4*c) + 2*(3*d*e - (3*c - 2*I)*f)*e^
(3*d*x + 3*c) - 2*(3*I*d*e + (-3*I*c - 2)*f)*e^(2*d*x + 2*c) - (3*d*e - (3*c - 2*I)*f)*e^(d*x + c))*log(e^(d*x
+ c) - 1) + 3*(-I*d*f*x - I*c*f + (d*f*x + c*f)*e^(5*d*x + 5*c) + (-I*d*f*x - I*c*f)*e^(4*d*x + 4*c) - 2*(d*f
*x + c*f)*e^(3*d*x + 3*c) + 2*(I*d*f*x + I*c*f)*e^(2*d*x + 2*c) + (d*f*x + c*f)*e^(d*x + c))*log(-e^(d*x + c)
+ 1))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3*c) + 2*I*a*d^2*e^(2*d*x + 2*c) +
a*d^2*e^(d*x + c) - I*a*d^2)

Sympy [F]

\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]

[In]

integrate((f*x+e)*csch(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*(Integral(e*csch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f*x*csch(c + d*x)**3/(sinh(c + d*x) - I), x
))/a

Maxima [F]

\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

-(24*d*integrate(1/16*x/(a*d*e^(d*x + c) + a*d), x) + 24*d*integrate(1/16*x/(a*d*e^(d*x + c) - a*d), x) + 8*(2
*d*x*e^(5*d*x + 5*c) + 2*I*d*x + (I*d*x*e^(4*c) + I*e^(4*c))*e^(4*d*x) - (d*x*e^(3*c) - e^(3*c))*e^(3*d*x) + (
-I*d*x*e^(2*c) - I*e^(2*c))*e^(2*d*x) + (d*x*e^c - e^c)*e^(d*x))/(8*I*a*d^2*e^(5*d*x + 5*c) + 8*a*d^2*e^(4*d*x
+ 4*c) - 16*I*a*d^2*e^(3*d*x + 3*c) - 16*a*d^2*e^(2*d*x + 2*c) + 8*I*a*d^2*e^(d*x + c) + 8*a*d^2) - 2*I*(d*x
+ c)/(a*d^2) + 2*I*log((e^(d*x + c) - I)*e^(-c))/(a*d^2) + I*log(e^(d*x + c) + 1)/(a*d^2) + I*log(e^(d*x + c)
- 1)/(a*d^2))*f - 1/2*e*(2*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) +
4)/((a*e^(-d*x - c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c) + I*a*e^(-4*d*x - 4*c) + a*e^(-5*d*x - 5*
c) + I*a)*d) - 3*log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d))

Giac [F]

\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((f*x+e)*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*csch(d*x + c)^3/(I*a*sinh(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)