3.2.41 \(\int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx\) [141]

Optimal. Leaf size=807 \[ \frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (4,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {PolyLog}\left (4,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}} \]

[Out]

3*x^2/a/f^2/(a+I*a*sinh(f*x+e))^(1/2)+24*I*x*arctanh(exp(1/2*e+3/4*I*Pi+1/2*f*x))*cosh(1/2*e+1/4*I*Pi+1/2*f*x)
/a/f^3/(a+I*a*sinh(f*x+e))^(1/2)-I*x^3*arctanh(exp(1/2*e+3/4*I*Pi+1/2*f*x))*cosh(1/2*e+1/4*I*Pi+1/2*f*x)/a/f/(
a+I*a*sinh(f*x+e))^(1/2)-24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a
*sinh(f*x+e))^(1/2)+3*I*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^2/(a+I*a*s
inh(f*x+e))^(1/2)+24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(
f*x+e))^(1/2)-3*I*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^2/(a+I*a*sinh(f
*x+e))^(1/2)-12*I*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(3,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^3/(a+I*a*sinh(f*x+
e))^(1/2)+12*I*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(3,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^3/(a+I*a*sinh(f*x+e)
)^(1/2)+24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(4,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(f*x+e))^(1/
2)-24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(4,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(f*x+e))^(1/2)+1
/2*x^3*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a/f/(a+I*a*sinh(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.34, antiderivative size = 807, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3400, 4271, 4267, 2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} \frac {\tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{2 a f \sqrt {i \sinh (e+f x) a+a}}+\frac {i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{a f \sqrt {i \sinh (e+f x) a+a}}+\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}+\frac {3 x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}+\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_4\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_4\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + I*a*Sinh[e + f*x])^(3/2),x]

[Out]

(3*x^2)/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*x*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 + (I/4)
*Pi + (f*x)/2])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x^3*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 +
(I/4)*Pi + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[2, -E^
((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) + ((3*I)*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*P
olyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi +
(f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - ((3*I)*x^2*Cosh[e/2 +
(I/4)*Pi + (f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((12*I)*x
*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]])
+ ((12*I)*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh
[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a
+ I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^
4*Sqrt[a + I*a*Sinh[e + f*x]]) + (x^3*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])

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 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 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 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 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*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
+ 1/2] && (GtQ[n, 0] || 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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

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 {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx &=-\frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \int x^3 \text {csch}^3\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{2 a \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \int x^3 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{4 a \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (3 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x^2 \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (3 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x^2 \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{2 a f \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {Li}_2\left (-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {Li}_2\left (e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \text {Li}_3\left (-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \text {Li}_3\left (e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_4\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_4\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.78, size = 546, normalized size = 0.68 \begin {gather*} \frac {\left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left (f^2 x^2 (6+i f x) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) (-1)^{3/4} \left (-48 e \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+2 e^3 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 e \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+e^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 e \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-e^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-f^3 x^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-6 \left (-8+f^2 x^2\right ) \text {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+6 \left (-8+f^2 x^2\right ) \text {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \text {PolyLog}\left (3,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \text {PolyLog}\left (3,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-48 \text {PolyLog}\left (4,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+48 \text {PolyLog}\left (4,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2+2 f^3 x^3 \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{2 f^4 (a+i a \sinh (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + I*a*Sinh[e + f*x])^(3/2),x]

[Out]

((Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])*(f^2*x^2*(6 + I*f*x)*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]) + (1
/2 - I/2)*(-1)^(3/4)*(-48*e*ArcTanh[(-1)^(3/4)*E^((e + f*x)/2)] + 2*e^3*ArcTanh[(-1)^(3/4)*E^((e + f*x)/2)] -
24*e*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + e^3*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] - 24*f*x*Log[1 - (-1)^(3/4)
*E^((e + f*x)/2)] + f^3*x^3*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + 24*e*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - e
^3*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - f^3*x^3*Log[1 + (-1)^(3/
4)*E^((e + f*x)/2)] - 6*(-8 + f^2*x^2)*PolyLog[2, -((-1)^(3/4)*E^((e + f*x)/2))] + 6*(-8 + f^2*x^2)*PolyLog[2,
(-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*PolyLog[3, -((-1)^(3/4)*E^((e + f*x)/2))] - 24*f*x*PolyLog[3, (-1)^(3/4)
*E^((e + f*x)/2)] - 48*PolyLog[4, -((-1)^(3/4)*E^((e + f*x)/2))] + 48*PolyLog[4, (-1)^(3/4)*E^((e + f*x)/2)])*
(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^2 + 2*f^3*x^3*Sinh[(e + f*x)/2]))/(2*f^4*(a + I*a*Sinh[e + f*x])^(3/
2))

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)

[Out]

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3/(I*a*sinh(f*x + e) + a)^(3/2), x)

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Fricas [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(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

((a^2*f^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*e^(f*x + e) - a^2*f^2)*integral(1/2*(-I*f^2*x^3 + 24*I*x)*sqrt(1/2*I*a
*e^(-f*x - e))*e^(f*x + e)/(a^2*f^2*e^(f*x + e) - I*a^2*f^2), x) + ((-I*f*x^3 - 6*I*x^2)*e^(2*f*x + 2*e) + (f*
x^3 - 6*x^2)*e^(f*x + e))*sqrt(1/2*I*a*e^(-f*x - e)))/(a^2*f^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*e^(f*x + e) - a^2
*f^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+I*a*sinh(f*x+e))**(3/2),x)

[Out]

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

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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(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(x^3/(I*a*sinh(f*x + e) + a)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + a*sinh(e + f*x)*1i)^(3/2),x)

[Out]

int(x^3/(a + a*sinh(e + f*x)*1i)^(3/2), x)

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