\(\int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx\) [146]

Optimal result

Integrand size = 21, antiderivative size = 1016 \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=-\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}} \]

[Out]

-1/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)+9/8*x^2/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)-10*I*cosh(1/2*c+1/4*I*Pi+1/2*d*
x)*polylog(2,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)+10*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*
polylog(2,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)-9/2*I*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)
*polylog(3,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)-9*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*pol
ylog(4,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)-9/8*I*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*
polylog(2,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)+9*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*pol
ylog(4,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)+10*I*x*arctanh(exp(1/2*c+3/4*I*Pi+1/2*d*
x))*cosh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)+9/2*I*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*polylo
g(3,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)+9/8*I*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*pol
ylog(2,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)-3/8*I*x^3*arctanh(exp(1/2*c+3/4*I*Pi+1/2
*d*x))*cosh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/2)+1/4*x^2*sech(1/2*c+1/4*I*Pi+1/2*d*x)^2/a^2
/d^2/(a+I*a*sinh(d*x+c))^(1/2)-1/2*x*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)+3/16*x^3*t
anh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/2)+1/8*x^3*sech(1/2*c+1/4*I*Pi+1/2*d*x)^2*tanh(1/2*c+
1/4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 1016, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3400, 4271, 4270, 4267, 2317, 2438, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\frac {\text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^3}{8 a^2 d \sqrt {i \sinh (c+d x) a+a}}+\frac {3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^3}{16 a^2 d \sqrt {i \sinh (c+d x) a+a}}+\frac {3 i \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^3}{8 a^2 d \sqrt {i \sinh (c+d x) a+a}}+\frac {\text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^2}{4 a^2 d^2 \sqrt {i \sinh (c+d x) a+a}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) x^2}{8 a^2 d^2 \sqrt {i \sinh (c+d x) a+a}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) x^2}{8 a^2 d^2 \sqrt {i \sinh (c+d x) a+a}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {i \sinh (c+d x) a+a}}-\frac {\tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x}{2 a^2 d^3 \sqrt {i \sinh (c+d x) a+a}}-\frac {10 i \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x}{a^2 d^3 \sqrt {i \sinh (c+d x) a+a}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) x}{2 a^2 d^3 \sqrt {i \sinh (c+d x) a+a}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) x}{2 a^2 d^3 \sqrt {i \sinh (c+d x) a+a}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {i \sinh (c+d x) a+a}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {i \sinh (c+d x) a+a}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {i \sinh (c+d x) a+a}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {i \sinh (c+d x) a+a}}-\frac {1}{a^2 d^4 \sqrt {i \sinh (c+d x) a+a}} \]

[In]

Int[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]

[Out]

-(1/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])) + (9*x^2)/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - ((10*I)*x*ArcTa
nh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((3*
I)/8)*x^3*ArcTanh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d*Sqrt[a + I*a*Sinh[c + d
*x]]) - ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*
a*Sinh[c + d*x]]) + (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(
a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^((2*c - I*Pi)/4 + (
d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^(
(2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/2)*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2
]*PolyLog[3, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((9*I)/2)*x*Cosh[c/2 + (
I/4)*Pi + (d*x)/2]*PolyLog[3, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + ((9*I)*Co
sh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])
- ((9*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c
+ d*x]]) + (x^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2)/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (x*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(16*a^2*d*S
qrt[a + I*a*Sinh[c + d*x]]) + (x^3*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(8*a^2*d*S
qrt[a + I*a*Sinh[c + d*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 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 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*} \text {integral}& = \frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int x^3 \text {csch}^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{4 a^2 \sqrt {a+i a \sinh (c+d x)}} \\ & = \frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^3 \text {csch}^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{16 a^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int x \text {csch}^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{2 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^3 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{32 a^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int x \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int \log \left (1-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int \log \left (1+e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \log \left (1-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \log \left (1+e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (9 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}} \\ & = -\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.99 (sec) , antiderivative size = 1200, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (-48 \cosh \left (\frac {1}{2} (c+d x)\right )+8 i c \cosh \left (\frac {1}{2} (c+d x)\right )+70 c^2 \cosh \left (\frac {1}{2} (c+d x)\right )-11 i c^3 \cosh \left (\frac {1}{2} (c+d x)\right )-8 i (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )-140 c (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )+33 i c^2 (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )+70 (c+d x)^2 \cosh \left (\frac {1}{2} (c+d x)\right )-33 i c (c+d x)^2 \cosh \left (\frac {1}{2} (c+d x)\right )+11 i (c+d x)^3 \cosh \left (\frac {1}{2} (c+d x)\right )+16 \cosh \left (\frac {3}{2} (c+d x)\right )+8 i c \cosh \left (\frac {3}{2} (c+d x)\right )-18 c^2 \cosh \left (\frac {3}{2} (c+d x)\right )-3 i c^3 \cosh \left (\frac {3}{2} (c+d x)\right )-8 i (c+d x) \cosh \left (\frac {3}{2} (c+d x)\right )+36 c (c+d x) \cosh \left (\frac {3}{2} (c+d x)\right )+9 i c^2 (c+d x) \cosh \left (\frac {3}{2} (c+d x)\right )-18 (c+d x)^2 \cosh \left (\frac {3}{2} (c+d x)\right )-9 i c (c+d x)^2 \cosh \left (\frac {3}{2} (c+d x)\right )+3 i (c+d x)^3 \cosh \left (\frac {3}{2} (c+d x)\right )+(1-i) (-1)^{3/4} \left (-160 c \text {arctanh}\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+6 c^3 \text {arctanh}\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-80 c \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+3 c^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-80 d x \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+3 d^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+80 c \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-3 c^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+80 d x \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-3 d^3 x^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-2 \left (-80+9 d^2 x^2\right ) \operatorname {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+2 \left (-80+9 d^2 x^2\right ) \operatorname {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+72 d x \operatorname {PolyLog}\left (3,-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-72 d x \operatorname {PolyLog}\left (3,(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-144 \operatorname {PolyLog}\left (4,-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+144 \operatorname {PolyLog}\left (4,(-1)^{3/4} e^{\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 )^4-48 i \sinh \left (\frac {1}{2} (c+d x)\right )+8 c \sinh \left (\frac {1}{2} (c+d x)\right )+70 i c^2 \sinh \left (\frac {1}{2} (c+d x)\right )-11 c^3 \sinh \left (\frac {1}{2} (c+d x)\right )-8 (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )-140 i c (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )+33 c^2 (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )+70 i (c+d x)^2 \sinh \left (\frac {1}{2} (c+d x)\right )-33 c (c+d x)^2 \sinh \left (\frac {1}{2} (c+d x)\right )+11 (c+d x)^3 \sinh \left (\frac {1}{2} (c+d x)\right )-16 i \sinh \left (\frac {3}{2} (c+d x)\right )-8 c \sinh \left (\frac {3}{2} (c+d x)\right )+18 i c^2 \sinh \left (\frac {3}{2} (c+d x)\right )+3 c^3 \sinh \left (\frac {3}{2} (c+d x)\right )+8 (c+d x) \sinh \left (\frac {3}{2} (c+d x)\right )-36 i c (c+d x) \sinh \left (\frac {3}{2} (c+d x)\right )-9 c^2 (c+d x) \sinh \left (\frac {3}{2} (c+d x)\right )+18 i (c+d x)^2 \sinh \left (\frac {3}{2} (c+d x)\right )+9 c (c+d x)^2 \sinh \left (\frac {3}{2} (c+d x)\right )-3 (c+d x)^3 \sinh \left (\frac {3}{2} (c+d x)\right )\right )}{32 d^4 (a+i a \sinh (c+d x))^{5/2}} \]

[In]

Integrate[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]

[Out]

((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(-48*Cosh[(c + d*x)/2] + (8*I)*c*Cosh[(c + d*x)/2] + 70*c^2*Cosh[(c
+ d*x)/2] - (11*I)*c^3*Cosh[(c + d*x)/2] - (8*I)*(c + d*x)*Cosh[(c + d*x)/2] - 140*c*(c + d*x)*Cosh[(c + d*x)
/2] + (33*I)*c^2*(c + d*x)*Cosh[(c + d*x)/2] + 70*(c + d*x)^2*Cosh[(c + d*x)/2] - (33*I)*c*(c + d*x)^2*Cosh[(c
+ d*x)/2] + (11*I)*(c + d*x)^3*Cosh[(c + d*x)/2] + 16*Cosh[(3*(c + d*x))/2] + (8*I)*c*Cosh[(3*(c + d*x))/2] -
18*c^2*Cosh[(3*(c + d*x))/2] - (3*I)*c^3*Cosh[(3*(c + d*x))/2] - (8*I)*(c + d*x)*Cosh[(3*(c + d*x))/2] + 36*c
*(c + d*x)*Cosh[(3*(c + d*x))/2] + (9*I)*c^2*(c + d*x)*Cosh[(3*(c + d*x))/2] - 18*(c + d*x)^2*Cosh[(3*(c + d*x
))/2] - (9*I)*c*(c + d*x)^2*Cosh[(3*(c + d*x))/2] + (3*I)*(c + d*x)^3*Cosh[(3*(c + d*x))/2] + (1 - I)*(-1)^(3/
4)*(-160*c*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2)] + 6*c^3*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2)] - 80*c*Log[1 - (-1)
^(3/4)*E^((c + d*x)/2)] + 3*c^3*Log[1 - (-1)^(3/4)*E^((c + d*x)/2)] - 80*d*x*Log[1 - (-1)^(3/4)*E^((c + d*x)/2
)] + 3*d^3*x^3*Log[1 - (-1)^(3/4)*E^((c + d*x)/2)] + 80*c*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*c^3*Log[1 +
(-1)^(3/4)*E^((c + d*x)/2)] + 80*d*x*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*d^3*x^3*Log[1 + (-1)^(3/4)*E^((c
+ d*x)/2)] - 2*(-80 + 9*d^2*x^2)*PolyLog[2, -((-1)^(3/4)*E^((c + d*x)/2))] + 2*(-80 + 9*d^2*x^2)*PolyLog[2, (-
1)^(3/4)*E^((c + d*x)/2)] + 72*d*x*PolyLog[3, -((-1)^(3/4)*E^((c + d*x)/2))] - 72*d*x*PolyLog[3, (-1)^(3/4)*E^
((c + d*x)/2)] - 144*PolyLog[4, -((-1)^(3/4)*E^((c + d*x)/2))] + 144*PolyLog[4, (-1)^(3/4)*E^((c + d*x)/2)])*(
Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^4 - (48*I)*Sinh[(c + d*x)/2] + 8*c*Sinh[(c + d*x)/2] + (70*I)*c^2*Sin
h[(c + d*x)/2] - 11*c^3*Sinh[(c + d*x)/2] - 8*(c + d*x)*Sinh[(c + d*x)/2] - (140*I)*c*(c + d*x)*Sinh[(c + d*x)
/2] + 33*c^2*(c + d*x)*Sinh[(c + d*x)/2] + (70*I)*(c + d*x)^2*Sinh[(c + d*x)/2] - 33*c*(c + d*x)^2*Sinh[(c + d
*x)/2] + 11*(c + d*x)^3*Sinh[(c + d*x)/2] - (16*I)*Sinh[(3*(c + d*x))/2] - 8*c*Sinh[(3*(c + d*x))/2] + (18*I)*
c^2*Sinh[(3*(c + d*x))/2] + 3*c^3*Sinh[(3*(c + d*x))/2] + 8*(c + d*x)*Sinh[(3*(c + d*x))/2] - (36*I)*c*(c + d*
x)*Sinh[(3*(c + d*x))/2] - 9*c^2*(c + d*x)*Sinh[(3*(c + d*x))/2] + (18*I)*(c + d*x)^2*Sinh[(3*(c + d*x))/2] +
9*c*(c + d*x)^2*Sinh[(3*(c + d*x))/2] - 3*(c + d*x)^3*Sinh[(3*(c + d*x))/2]))/(32*d^4*(a + I*a*Sinh[c + d*x])^
(5/2))

Maple [F]

\[\int \frac {x^{3}}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

[In]

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)

[Out]

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)

Fricas [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/8*(8*(a^3*d^4*e^(4*d*x + 4*c) - 4*I*a^3*d^4*e^(3*d*x + 3*c) - 6*a^3*d^4*e^(2*d*x + 2*c) + 4*I*a^3*d^4*e^(d*x
+ c) + a^3*d^4)*integral(1/16*(-3*I*d^2*x^3 + 80*I*x)*sqrt(1/2*I*a*e^(-d*x - c))*e^(d*x + c)/(a^3*d^2*e^(d*x
+ c) - I*a^3*d^2), x) + ((-3*I*d^3*x^3 - 18*I*d^2*x^2 + 8*I*d*x + 16*I)*e^(4*d*x + 4*c) - (11*d^3*x^3 + 70*d^2
*x^2 - 8*d*x - 48)*e^(3*d*x + 3*c) + (-11*I*d^3*x^3 + 70*I*d^2*x^2 + 8*I*d*x - 48*I)*e^(2*d*x + 2*c) - (3*d^3*
x^3 - 18*d^2*x^2 - 8*d*x + 16)*e^(d*x + c))*sqrt(1/2*I*a*e^(-d*x - c)))/(a^3*d^4*e^(4*d*x + 4*c) - 4*I*a^3*d^4
*e^(3*d*x + 3*c) - 6*a^3*d^4*e^(2*d*x + 2*c) + 4*I*a^3*d^4*e^(d*x + c) + a^3*d^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(x**3/(a+I*a*sinh(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)

Giac [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int \frac {x^3}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]

[In]

int(x^3/(a + a*sinh(c + d*x)*1i)^(5/2),x)

[Out]

int(x^3/(a + a*sinh(c + d*x)*1i)^(5/2), x)