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

Optimal. Leaf size=214 \[ \frac{3 f \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac{3 f \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{f \text{csch}(c+d x)}{2 a d^2}-\frac{i f \log (\sinh (c+d x))}{a d^2}-\frac{2 i f \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )\right )}{a d^2}+\frac{3 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x) \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x) \coth (c+d x)}{a d}-\frac{(e+f x) \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

[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)

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Rubi [A]  time = 0.367409, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {5575, 4185, 4182, 2279, 2391, 4184, 3475, 3318} \[ \frac{3 f \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac{3 f \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{f \text{csch}(c+d x)}{2 a d^2}-\frac{i f \log (\sinh (c+d x))}{a d^2}-\frac{2 i f \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )\right )}{a d^2}+\frac{3 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x) \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x) \coth (c+d x)}{a d}-\frac{(e+f x) \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[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 5575

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]

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 4182

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 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 4184

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 3475

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

Rule 3318

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*(e + (Pi*a)/(2*b)))/2 + (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])

Rubi steps

\begin{align*} \int \frac{(e+f x) \text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\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) \tanh ^{-1}\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) \tanh ^{-1}\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 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac{f \operatorname{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) \tanh ^{-1}\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 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{f \text{Li}_2\left (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 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}-\frac{f \operatorname{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) \tanh ^{-1}\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 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 f \text{Li}_2\left (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]  time = 2.88705, size = 541, normalized size = 2.53 \[ \frac{\left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right ) \left (-12 f \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right ) \left (\text{PolyLog}\left (2,-e^{-c-d x}\right )-\text{PolyLog}\left (2,e^{-c-d x}\right )+(c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )+16 i d (e+f x) \sinh \left (\frac{1}{2} (c+d x)\right )+2 i \cosh \left (\frac{1}{2} (c+d x)\right ) \left (\coth \left (\frac{1}{2} (c+d x)\right )+i\right ) (2 d (e+f x)+i f)-d (e+f x) \left (\coth \left (\frac{1}{2} (c+d x)\right )+i\right ) \text{csch}\left (\frac{1}{2} (c+d x)\right )-i d (e+f x) \left (\tanh \left (\frac{1}{2} (c+d x)\right )-i\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right )+2 \tanh \left (\frac{1}{2} (c+d x)\right ) (f+2 i d (e+f x)) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )-12 d e \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )-8 f (c+d x) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )+8 f \log (\cosh (c+d x)) \left (\sinh \left (\frac{1}{2} (c+d x)\right )-i \cosh \left (\frac{1}{2} (c+d x)\right )\right )+8 f \log (\sinh (c+d x)) \left (\sinh \left (\frac{1}{2} (c+d x)\right )-i \cosh \left (\frac{1}{2} (c+d x)\right )\right )+12 c f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )+16 f \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{8 d^2 (a+i a \sinh (c+d x))} \]

Antiderivative was successfully verified.

[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]) - 12*d*e*Log[Tanh[(c + d*x)
/2]]*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + 12*c*f*Log[Tanh[(c + d*x)/2]]*(Cosh[(c + d*x)/2] + I*Sinh[(c
+ d*x)/2]) - 12*f*((c + d*x)*(Log[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + PolyLog[2, -E^(-c - d*x)] - Pol
yLog[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]) + 8*f*Log[Sinh[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]))

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Maple [B]  time = 0.184, size = 423, normalized size = 2. \begin{align*} -{\frac{-3\,idfx{{\rm e}^{3\,dx+3\,c}}-5\,dfx{{\rm e}^{2\,dx+2\,c}}+3\,dfx{{\rm e}^{4\,dx+4\,c}}+idfx{{\rm e}^{dx+c}}-5\,de{{\rm e}^{2\,dx+2\,c}}+3\,de{{\rm e}^{4\,dx+4\,c}}-3\,ide{{\rm e}^{3\,dx+3\,c}}+4\,dfx+f{{\rm e}^{4\,dx+4\,c}}+i{{\rm e}^{dx+c}}f-i{{\rm e}^{3\,dx+3\,c}}f+4\,de-f{{\rm e}^{2\,dx+2\,c}}+ide{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{2}{d}^{2} \left ({{\rm e}^{dx+c}}-i \right ) a}}-{\frac{3\,f{\it polylog} \left ( 2,{{\rm e}^{dx+c}} \right ) }{2\,a{d}^{2}}}+{\frac{3\,f{\it polylog} \left ( 2,-{{\rm e}^{dx+c}} \right ) }{2\,a{d}^{2}}}-{\frac{2\,if\ln \left ({{\rm e}^{dx+c}}-i \right ) }{a{d}^{2}}}+{\frac{4\,if\ln \left ({{\rm e}^{dx+c}} \right ) }{a{d}^{2}}}-{\frac{if\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{a{d}^{2}}}-{\frac{if\ln \left ({{\rm e}^{dx+c}}+1 \right ) }{a{d}^{2}}}-{\frac{3\,e\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{2\,da}}+{\frac{3\,e\ln \left ({{\rm e}^{dx+c}}+1 \right ) }{2\,da}}-{\frac{3\,\ln \left ( 1-{{\rm e}^{dx+c}} \right ) fx}{2\,da}}-{\frac{3\,\ln \left ( 1-{{\rm e}^{dx+c}} \right ) cf}{2\,a{d}^{2}}}+{\frac{3\,f\ln \left ({{\rm e}^{dx+c}}+1 \right ) x}{2\,da}}+{\frac{3\,fc\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{2\,a{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-3*I*d*f*x*exp(3*d*x+3*c)-5*d*f*x*exp(2*d*x+2*c)+3*d*f*x*exp(4*d*x+4*c)+I*d*f*x*exp(d*x+c)-5*d*e*exp(2*d*x+2
*c)+3*d*e*exp(4*d*x+4*c)-3*I*d*e*exp(3*d*x+3*c)+4*d*f*x+f*exp(4*d*x+4*c)+I*exp(d*x+c)*f-I*exp(3*d*x+3*c)*f+4*d
*e-f*exp(2*d*x+2*c)+I*d*e*exp(d*x+c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/a-3/2*f*polylog(2,exp(d*x+c))/a/
d^2+3/2*f*polylog(2,-exp(d*x+c))/a/d^2-2*I*f/a/d^2*ln(exp(d*x+c)-I)+4*I/d^2/a*f*ln(exp(d*x+c))-I/d^2/a*f*ln(ex
p(d*x+c)-1)-I/d^2/a*f*ln(exp(d*x+c)+1)-3/2/d/a*e*ln(exp(d*x+c)-1)+3/2/d/a*e*ln(exp(d*x+c)+1)-3/2/d/a*ln(1-exp(
d*x+c))*f*x-3/2/d^2/a*ln(1-exp(d*x+c))*c*f+3/2/d/a*ln(exp(d*x+c)+1)*f*x+3/2/d^2/a*f*c*ln(exp(d*x+c)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (24 \, d \int \frac{x}{16 \,{\left (a d e^{\left (d x + c\right )} + a d\right )}}\,{d x} + 24 \, d \int \frac{x}{16 \,{\left (a d e^{\left (d x + c\right )} - a d\right )}}\,{d x} + \frac{8 \,{\left (2 \, d x e^{\left (5 \, d x + 5 \, c\right )} + 2 i \, d x +{\left (i \, d x e^{\left (4 \, c\right )} + i \, e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} -{\left (d x e^{\left (3 \, c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (-i \, d x e^{\left (2 \, c\right )} - i \, e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} +{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}\right )}}{8 i \, a d^{2} e^{\left (5 \, d x + 5 \, c\right )} + 8 \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 i \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 16 \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 i \, a d^{2} e^{\left (d x + c\right )} + 8 \, a d^{2}} - \frac{2 i \,{\left (d x + c\right )}}{a d^{2}} + \frac{2 i \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}} + \frac{i \, \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac{i \, \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}}\right )} f - \frac{1}{2} \, e{\left (\frac{16 \,{\left (-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4\right )}}{{\left (8 \, a e^{\left (-d x - c\right )} - 16 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 16 \, a e^{\left (-3 \, d x - 3 \, c\right )} + 8 i \, a e^{\left (-4 \, d x - 4 \, c\right )} + 8 \, a e^{\left (-5 \, d x - 5 \, c\right )} + 8 i \, a\right )} d} - \frac{3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac{3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [B]  time = 2.77223, size = 2163, normalized size = 10.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*d*e - 4*c*f - (3*f*e^(5*d*x + 5*c) - 3*I*f*e^(4*d*x + 4*c) - 6*f*e^(3*d*x + 3*c) + 6*I*f*e^(2*d*x + 2*c) +
3*f*e^(d*x + c) - 3*I*f)*dilog(-e^(d*x + c)) + (3*f*e^(5*d*x + 5*c) - 3*I*f*e^(4*d*x + 4*c) - 6*f*e^(3*d*x +
3*c) + 6*I*f*e^(2*d*x + 2*c) + 3*f*e^(d*x + c) - 3*I*f)*dilog(e^(d*x + c)) - (8*I*d*f*x + 4*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) - (-10*I*d*f*x + 6*I*d*e + (-8*I*c + 2*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) - (6*I*d*f*x - 2*I*d*e + (4*I*c - 2*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) + (6*I*d*f*x + 6*I*d*e + 4*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) - 4*f*e^(4*d*x + 4*c) + 8*I
*f*e^(3*d*x + 3*c) + 8*f*e^(2*d*x + 2*c) - 4*I*f*e^(d*x + c) - 4*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) + (6*d*e - (6*c
- 4*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 + 3*I*c*f - 3*(d*f*x + c*f)*e^(5*d*x + 5*c) + (3*I*d*f*x + 3*I*c*f)*e^(4*d
*x + 4*c) + 6*(d*f*x + c*f)*e^(3*d*x + 3*c) + (-6*I*d*f*x - 6*I*c*f)*e^(2*d*x + 2*c) - 3*(d*f*x + c*f)*e^(d*x
+ c))*log(-e^(d*x + c) + 1))/(2*a*d^2*e^(5*d*x + 5*c) - 2*I*a*d^2*e^(4*d*x + 4*c) - 4*a*d^2*e^(3*d*x + 3*c) +
4*I*a*d^2*e^(2*d*x + 2*c) + 2*a*d^2*e^(d*x + c) - 2*I*a*d^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)