3.3.5 \(\int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [205]

Optimal. Leaf size=313 \[ -\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

-I*(f*x+e)^3/a/d-2*(f*x+e)^3*arctanh(exp(d*x+c))/a/d+6*I*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2-3*f*(f*x+e)^2*po
lylog(2,-exp(d*x+c))/a/d^2+12*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+3*f*(f*x+e)^2*polylog(2,exp(d*x+c))
/a/d^2+6*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-6*f^2*(f*x+e)*polylo
g(3,exp(d*x+c))/a/d^3-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*polylog(4,exp(d*x+c))/a/d^4-I*(f*x+e)^3*tanh(1/
2*c+1/4*I*Pi+1/2*d*x)/a/d

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Rubi [A]
time = 0.36, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5694, 4267, 2611, 6744, 2320, 6724, 3399, 4269, 3797, 2221} \begin {gather*} -\frac {12 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^3}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(e + f*x)^3)/(a*d) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) + ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c +
d*x)])/(a*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^
(c + d*x)])/(a*d^3) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -E^(c +
d*x)])/(a*d^3) - ((12*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])
/(a*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (I*(e + f*x)^3
*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{a}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {i \int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 i f) \int (e+f x)^2 \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}+\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(6 f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^3\right ) \int \text {Li}_2\left (-i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}\\ \end {align*}

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Mathematica [A]
time = 4.41, size = 501, normalized size = 1.60 \begin {gather*} \frac {-2 d^3 e^3 \tanh ^{-1}\left (e^{c+d x}\right )+3 d^3 e^2 f x \log \left (1-e^{c+d x}\right )+3 d^3 e f^2 x^2 \log \left (1-e^{c+d x}\right )+d^3 f^3 x^3 \log \left (1-e^{c+d x}\right )-3 d^3 e^2 f x \log \left (1+e^{c+d x}\right )-3 d^3 e f^2 x^2 \log \left (1+e^{c+d x}\right )-d^3 f^3 x^3 \log \left (1+e^{c+d x}\right )-3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )+3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )+6 d e f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+6 d f^3 x \text {PolyLog}\left (3,-e^{c+d x}\right )+\frac {2 f \left (d^2 \left (-i d e^c x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (1+i e^c\right ) (e+f x)^2 \log \left (1+i e^{c+d x}\right )\right )+6 d \left (1+i e^c\right ) f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )-6 i \left (-i+e^c\right ) f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )\right )}{-i+e^c}-6 d e f^2 \text {PolyLog}\left (3,e^{c+d x}\right )-6 d f^3 x \text {PolyLog}\left (3,e^{c+d x}\right )-6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )+6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )-\frac {2 i d^3 (e+f x)^3 \sinh \left (\frac {d x}{2}\right )}{\left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*e^3*ArcTanh[E^(c + d*x)] + 3*d^3*e^2*f*x*Log[1 - E^(c + d*x)] + 3*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] +
d^3*f^3*x^3*Log[1 - E^(c + d*x)] - 3*d^3*e^2*f*x*Log[1 + E^(c + d*x)] - 3*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)]
- d^3*f^3*x^3*Log[1 + E^(c + d*x)] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)] + 3*d^2*f*(e + f*x)^2*PolyLo
g[2, E^(c + d*x)] + 6*d*e*f^2*PolyLog[3, -E^(c + d*x)] + 6*d*f^3*x*PolyLog[3, -E^(c + d*x)] + (2*f*(d^2*((-I)*
d*E^c*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 3*(1 + I*E^c)*(e + f*x)^2*Log[1 + I*E^(c + d*x)]) + 6*d*(1 + I*E^c)*f*(e
+ f*x)*PolyLog[2, (-I)*E^(c + d*x)] - (6*I)*(-I + E^c)*f^2*PolyLog[3, (-I)*E^(c + d*x)]))/(-I + E^c) - 6*d*e*
f^2*PolyLog[3, E^(c + d*x)] - 6*d*f^3*x*PolyLog[3, E^(c + d*x)] - 6*f^3*PolyLog[4, -E^(c + d*x)] + 6*f^3*PolyL
og[4, E^(c + d*x)] - ((2*I)*d^3*(e + f*x)^3*Sinh[(d*x)/2])/((Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*S
inh[(c + d*x)/2])))/(a*d^4)

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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. 1033 vs. \(2 (288 ) = 576\).
time = 3.34, size = 1034, normalized size = 3.30

method	result	size
risch	\(\frac {e^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}-\frac {e^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}-\frac {6 f^{3} \polylog \left (4, -{\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 f^{3} \polylog \left (4, {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {12 i f^{3} \polylog \left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {12 i e \,f^{2} c x}{a \,d^{2}}+\frac {3 e^{2} f \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {3 f^{3} \polylog \left (2, {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {6 f^{3} \polylog \left (3, {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}-\frac {6 i e \,f^{2} c^{2}}{a \,d^{3}}-\frac {6 i f^{3} c^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) c \,e^{2} f}{a \,d^{2}}-\frac {3 e \,f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{2}}{a d}+\frac {3 e \,f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{2}}{a d}-\frac {3 e \,f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{3}}+\frac {3 e \,f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) e^{2} f x}{a d}+\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) e^{2} f x}{a d}-\frac {3 e^{2} f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {12 i e \,f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {6 e \,f^{2} \polylog \left (2, -{\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {6 e \,f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {6 i \ln \left ({\mathrm e}^{d x +c}\right ) e^{2} f}{a \,d^{2}}+\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {6 i f^{3} c^{2} x}{a \,d^{3}}+\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {2 f^{3} x^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2} f}{a \,d^{2}}+\frac {12 i e \,f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {6 i e \,f^{2} x^{2}}{a d}+\frac {12 i e \,f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i f^{3} \polylog \left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}-\frac {2 i f^{3} x^{3}}{a d}+\frac {6 e \,f^{2} \polylog \left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {6 e \,f^{2} \polylog \left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {3 f^{3} \polylog \left (2, -{\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {6 f^{3} \polylog \left (3, -{\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}-\frac {f^{3} c^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{4}}-\frac {f^{3} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{3}}{a d}+\frac {f^{3} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{3}}{a d}+\frac {f^{3} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{3}}{a \,d^{4}}+\frac {4 i f^{3} c^{3}}{a \,d^{4}}-\frac {3 e^{2} f \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}\)	\(1034\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*I/a/d^3*f^3*c^2*x+12*I/a/d^3*e*f^2*polylog(2,-I*exp(d*x+c))+6*I/a/d^4*f^3*c^2*ln(exp(d*x+c)-I)-6*I/a/d^4*f^3
*c^2*ln(exp(d*x+c))-6*I/a/d*e*f^2*x^2-6*I/a/d^3*e*f^2*c^2+1/a/d*e^3*ln(exp(d*x+c)-1)-1/a/d*e^3*ln(exp(d*x+c)+1
)+3/a/d^2*ln(1-exp(d*x+c))*c*e^2*f-3/a/d*e*f^2*ln(exp(d*x+c)+1)*x^2-6/a/d^2*e*f^2*polylog(2,-exp(d*x+c))*x+3/a
/d*e*f^2*ln(1-exp(d*x+c))*x^2-3/a/d^3*e*f^2*ln(1-exp(d*x+c))*c^2+6/a/d^2*e*f^2*polylog(2,exp(d*x+c))*x-6*I/a/d
^4*f^3*c^2*ln(1+I*exp(d*x+c))+6*I/a/d^2*f^3*ln(1+I*exp(d*x+c))*x^2+12*I/a/d^3*f^3*polylog(2,-I*exp(d*x+c))*x-6
*I/a/d^2*e^2*f*ln(exp(d*x+c))+6*I/a/d^2*e^2*f*ln(exp(d*x+c)-I)+12*I/a/d^2*e*f^2*ln(1+I*exp(d*x+c))*x+12*I/a/d^
3*e*f^2*c*ln(exp(d*x+c))-12*I/a/d^3*e*f^2*c*ln(exp(d*x+c)-I)+12*I/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c+3/a/d^3*e*f
^2*c^2*ln(exp(d*x+c)-1)-12*I/a/d^2*e*f^2*c*x+2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3)/d/a/(exp(d*x+c)-I)-3/a/d*ln
(exp(d*x+c)+1)*e^2*f*x+3/a/d*ln(1-exp(d*x+c))*e^2*f*x-3/a/d^2*e^2*f*c*ln(exp(d*x+c)-1)-12*I*f^3*polylog(3,-I*e
xp(d*x+c))/a/d^4-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*polylog(4,exp(d*x+c))/a/d^4-3/a/d^2*e^2*f*polylog(2,
-exp(d*x+c))+3/a/d^2*e^2*f*polylog(2,exp(d*x+c))+3/a/d^2*f^3*polylog(2,exp(d*x+c))*x^2-6/a/d^3*f^3*polylog(3,e
xp(d*x+c))*x-1/a/d^4*f^3*c^3*ln(exp(d*x+c)-1)+6/a/d^3*e*f^2*polylog(3,-exp(d*x+c))-6/a/d^3*e*f^2*polylog(3,exp
(d*x+c))-1/a/d*f^3*ln(exp(d*x+c)+1)*x^3-3/a/d^2*f^3*polylog(2,-exp(d*x+c))*x^2+6/a/d^3*f^3*polylog(3,-exp(d*x+
c))*x+1/a/d*f^3*ln(1-exp(d*x+c))*x^3+1/a/d^4*f^3*ln(1-exp(d*x+c))*c^3-2*I/a/d*f^3*x^3+4*I/a/d^4*f^3*c^3

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (285) = 570\).
time = 0.45, size = 579, normalized size = 1.85 \begin {gather*} -{\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d}\right )} e^{3} - \frac {6 i \, f x e^{2}}{a d} + \frac {2 \, {\left (f^{3} x^{3} + 3 \, f^{2} x^{2} e + 3 \, f x e^{2}\right )}}{a d e^{\left (d x + c\right )} - i \, a d} - \frac {3 \, {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} f e^{2}}{a d^{2}} + \frac {3 \, {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} f e^{2}}{a d^{2}} - \frac {3 \, {\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2} e}{a d^{3}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2} e}{a d^{3}} + \frac {12 i \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2} e}{a d^{3}} + \frac {6 i \, f e^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{2}} - \frac {{\left (d^{3} x^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {{\left (d^{3} x^{3} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {6 i \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {2 \, {\left (-i \, d^{3} f^{3} x^{3} - 3 i \, d^{3} f^{2} x^{2} e\right )}}{a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d) - 2/((a*e^(-d*x - c) + I*a)*d))*e^3 - 6*I*f*x*e^2/
(a*d) + 2*(f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2)/(a*d*e^(d*x + c) - I*a*d) - 3*(d*x*log(e^(d*x + c) + 1) + dilog(
-e^(d*x + c)))*f*e^2/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*f*e^2/(a*d^2) - 3*(d^2*x^2*l
og(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2*e/(a*d^3) + 3*(d^2*x^2*log(-
e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2*e/(a*d^3) + 12*I*(d*x*log(I*e^(d*
x + c) + 1) + dilog(-I*e^(d*x + c)))*f^2*e/(a*d^3) + 6*I*f*e^2*log(I*e^(d*x + c) + 1)/(a*d^2) - (d^3*x^3*log(e
^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))
*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) +
6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) + 6*I*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*dilog(-I*e^(d*x + c)) -
2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) + 2*(-I*d^3*f^3*x^3 - 3*I*d^3*f^2*x^2*e)/(a*d^4)

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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. 1013 vs. \(2 (285) = 570\).
time = 0.41, size = 1013, normalized size = 3.24 \begin {gather*} -\frac {2 \, c^{3} f^{3} - 6 \, c^{2} d f^{2} e + 6 \, c d^{2} f e^{2} - 2 \, d^{3} e^{3} - 12 \, {\left (d f^{3} x + d f^{2} e - {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 3 \, {\left (-i \, d^{2} f^{3} x^{2} - 2 i \, d^{2} f^{2} x e - i \, d^{2} f e^{2} + {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} f^{2} x e + d^{2} f e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) + 3 \, {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} f^{2} x e + i \, d^{2} f e^{2} - {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} f^{2} x e + d^{2} f e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) + 2 \, {\left (i \, d^{3} f^{3} x^{3} + i \, c^{3} f^{3} + 3 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} e^{2} + 3 \, {\left (i \, d^{3} f^{2} x^{2} - i \, c^{2} d f^{2}\right )} e\right )} e^{\left (d x + c\right )} - {\left (i \, d^{3} f^{3} x^{3} + 3 i \, d^{3} f^{2} x^{2} e + 3 i \, d^{3} f x e^{2} + i \, d^{3} e^{3} - {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} f^{2} x^{2} e + 3 \, d^{3} f x e^{2} + d^{3} e^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 6 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2} - {\left (-i \, c^{2} f^{3} + 2 i \, c d f^{2} e - i \, d^{2} f e^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - {\left (i \, c^{3} f^{3} - 3 i \, c^{2} d f^{2} e + 3 i \, c d^{2} f e^{2} - i \, d^{3} e^{3} - {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 6 \, {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e - {\left (-i \, d^{2} f^{3} x^{2} + i \, c^{2} f^{3} + 2 \, {\left (-i \, d^{2} f^{2} x - i \, c d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) - {\left (-i \, d^{3} f^{3} x^{3} - i \, c^{3} f^{3} - 3 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} e^{2} - 3 \, {\left (i \, d^{3} f^{2} x^{2} - i \, c^{2} d f^{2}\right )} e + {\left (d^{3} f^{3} x^{3} + c^{3} f^{3} + 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{2} d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 6 \, {\left (f^{3} e^{\left (d x + c\right )} - i \, f^{3}\right )} {\rm polylog}\left (4, -e^{\left (d x + c\right )}\right ) - 6 \, {\left (f^{3} e^{\left (d x + c\right )} - i \, f^{3}\right )} {\rm polylog}\left (4, e^{\left (d x + c\right )}\right ) + 12 \, {\left (i \, f^{3} e^{\left (d x + c\right )} + f^{3}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right ) + 6 \, {\left (i \, d f^{3} x + i \, d f^{2} e - {\left (d f^{3} x + d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -e^{\left (d x + c\right )}\right ) + 6 \, {\left (-i \, d f^{3} x - i \, d f^{2} e + {\left (d f^{3} x + d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, e^{\left (d x + c\right )}\right )}{a d^{4} e^{\left (d x + c\right )} - i \, a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c^3*f^3 - 6*c^2*d*f^2*e + 6*c*d^2*f*e^2 - 2*d^3*e^3 - 12*(d*f^3*x + d*f^2*e - (-I*d*f^3*x - I*d*f^2*e)*e^(
d*x + c))*dilog(-I*e^(d*x + c)) + 3*(-I*d^2*f^3*x^2 - 2*I*d^2*f^2*x*e - I*d^2*f*e^2 + (d^2*f^3*x^2 + 2*d^2*f^2
*x*e + d^2*f*e^2)*e^(d*x + c))*dilog(-e^(d*x + c)) + 3*(I*d^2*f^3*x^2 + 2*I*d^2*f^2*x*e + I*d^2*f*e^2 - (d^2*f
^3*x^2 + 2*d^2*f^2*x*e + d^2*f*e^2)*e^(d*x + c))*dilog(e^(d*x + c)) + 2*(I*d^3*f^3*x^3 + I*c^3*f^3 + 3*(I*d^3*
f*x + I*c*d^2*f)*e^2 + 3*(I*d^3*f^2*x^2 - I*c^2*d*f^2)*e)*e^(d*x + c) - (I*d^3*f^3*x^3 + 3*I*d^3*f^2*x^2*e + 3
*I*d^3*f*x*e^2 + I*d^3*e^3 - (d^3*f^3*x^3 + 3*d^3*f^2*x^2*e + 3*d^3*f*x*e^2 + d^3*e^3)*e^(d*x + c))*log(e^(d*x
+ c) + 1) - 6*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 - (-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*e^(d*x + c))*lo
g(e^(d*x + c) - I) - (I*c^3*f^3 - 3*I*c^2*d*f^2*e + 3*I*c*d^2*f*e^2 - I*d^3*e^3 - (c^3*f^3 - 3*c^2*d*f^2*e + 3
*c*d^2*f*e^2 - d^3*e^3)*e^(d*x + c))*log(e^(d*x + c) - 1) - 6*(d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)
*e - (-I*d^2*f^3*x^2 + I*c^2*f^3 + 2*(-I*d^2*f^2*x - I*c*d*f^2)*e)*e^(d*x + c))*log(I*e^(d*x + c) + 1) - (-I*d
^3*f^3*x^3 - I*c^3*f^3 - 3*(I*d^3*f*x + I*c*d^2*f)*e^2 - 3*(I*d^3*f^2*x^2 - I*c^2*d*f^2)*e + (d^3*f^3*x^3 + c^
3*f^3 + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*e)*e^(d*x + c))*log(-e^(d*x + c) + 1) + 6*(f^3
*e^(d*x + c) - I*f^3)*polylog(4, -e^(d*x + c)) - 6*(f^3*e^(d*x + c) - I*f^3)*polylog(4, e^(d*x + c)) + 12*(I*f
^3*e^(d*x + c) + f^3)*polylog(3, -I*e^(d*x + c)) + 6*(I*d*f^3*x + I*d*f^2*e - (d*f^3*x + d*f^2*e)*e^(d*x + c))
*polylog(3, -e^(d*x + c)) + 6*(-I*d*f^3*x - I*d*f^2*e + (d*f^3*x + d*f^2*e)*e^(d*x + c))*polylog(3, e^(d*x + c
)))/(a*d^4*e^(d*x + c) - I*a*d^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \left (\int \frac {e^{3} \operatorname {csch}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {csch}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {csch}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {csch}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(Integral(e**3*csch(c + d*x)/(sinh(c + d*x) - I), x) + Integral(f**3*x**3*csch(c + d*x)/(sinh(c + d*x) - I)
, x) + Integral(3*e*f**2*x**2*csch(c + d*x)/(sinh(c + d*x) - I), x) + Integral(3*e**2*f*x*csch(c + d*x)/(sinh(
c + d*x) - I), x))/a

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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