3.586 \(\int \frac{(f-i c f x)^{5/2} (a+b \sinh ^{-1}(c x))^2}{(d+i c d x)^{3/2}} \, dx\)

Optimal. Leaf size=972 \[ \text{result too large to display} \]

[Out]

((-8*I)*a*b*f^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b^2*f^4*(1 + c^2*x^2
)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b^2*f^4*x*(1 + c^2*x^2)^2)/(4*(d + I*c*d*x)^(3/2)*(f - I*c
*f*x)^(3/2)) - (b^2*f^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*
I)*b^2*f^4*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*c*f^4*x^2*(1 + c
^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*f^4*(1 + c^2*x^2)*(a
+ b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])
^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c
*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*f^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(
f - I*c*f*x)^(3/2)) + (f^4*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2
)) - (5*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((32
*I)*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x
)^(3/2)) - (16*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3
/2)*(f - I*c*f*x)^(3/2)) - (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(
3/2)*(f - I*c*f*x)^(3/2)) + (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/
2)*(f - I*c*f*x)^(3/2)) - (8*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/
2)*(f - I*c*f*x)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.36351, antiderivative size = 972, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 19, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.514, Rules used = {5712, 5833, 5821, 5687, 5714, 3718, 2190, 2279, 2391, 5717, 5693, 4180, 5675, 5653, 261, 5758, 5661, 321, 215} \[ -\frac{5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3 f^4}{2 b c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 x \left (c^2 x^2+1\right )^2 f^4}{4 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 \left (c^2 x^2+1\right )^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 f^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i a b x \left (c^2 x^2+1\right )^{3/2} f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 x \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) f^4}{4 c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b c x^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) f^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right ) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b^2 \left (c^2 x^2+1\right )^{3/2} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{16 b^2 \left (c^2 x^2+1\right )^{3/2} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 \left (c^2 x^2+1\right )^{3/2} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2),x]

[Out]

((-8*I)*a*b*f^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b^2*f^4*(1 + c^2*x^2
)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b^2*f^4*x*(1 + c^2*x^2)^2)/(4*(d + I*c*d*x)^(3/2)*(f - I*c
*f*x)^(3/2)) - (b^2*f^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*
I)*b^2*f^4*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*c*f^4*x^2*(1 + c
^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*f^4*(1 + c^2*x^2)*(a
+ b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])
^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c
*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*f^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(
f - I*c*f*x)^(3/2)) + (f^4*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2
)) - (5*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((32
*I)*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x
)^(3/2)) - (16*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3
/2)*(f - I*c*f*x)^(3/2)) - (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(
3/2)*(f - I*c*f*x)^(3/2)) + (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/
2)*(f - I*c*f*x)^(3/2)) - (8*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/
2)*(f - I*c*f*x)^(3/2))

Rule 5712

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :>
Dist[((d + e*x)^q*(f + g*x)^q)/(1 + c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n,
x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p,
q] && GeQ[p - q, 0]

Rule 5833

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 5821

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p,
-1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5687

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSinh
[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5714

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5693

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{(f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{3/2} \int \frac{(f-i c f x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{\left (1+c^2 x^2\right )^{3/2} \int \left (-\frac{8 i \left (i f^4+c f^4 x\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac{7 f^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{4 i c f^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{c^2 f^4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{\left (8 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{\left (i f^4+c f^4 x\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (7 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (4 i c f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (c^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{7 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (8 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (\frac{i f^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac{c f^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (8 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (b c f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (8 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (8 i b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \sinh ^{-1}(c x) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (8 i c f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (b^2 c^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 b c f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (8 i b^2 c f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 f^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 f^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (32 b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 f^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 f^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (8 b^2 f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{8 i a b f^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 f^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 f^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 f^4 x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i f^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f^4 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{5 f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b f^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{16 b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 f^4 \left (1+c^2 x^2\right )^{3/2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end{align*}

Mathematica [B]  time = 11.6129, size = 2492, normalized size = 2.56 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2),x]

[Out]

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((4*I)*a^2*f^2)/d^2 + (a^2*c*f^2*x)/(2*d^2) + (8*a^2*f^2)/(d^2*(
-I + c*x))))/c - (15*a^2*f^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(
2*c*d^(3/2)) + ((4*I)*a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Co
sh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Cot
h[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]]) + I*(-(c*x) - 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*
x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/
(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2
])) - (a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]
/2]*(ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]]) + I*(
ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSin
h[c*x]/2]))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[Ar
cSinh[c*x]/2])) - (b^2*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[
ArcSinh[c*x]/2]*((6*I)*Pi*ArcSinh[c*x] + (6 - 6*I)*ArcSinh[c*x]^2 + ArcSinh[c*x]^3 + 12*((-I)*Pi + 2*ArcSinh[c
*x])*Log[1 - I/E^ArcSinh[c*x]] - (24*I)*Pi*Log[1 + E^ArcSinh[c*x]] + (24*I)*Pi*Log[Cosh[ArcSinh[c*x]/2]] + (12
*I)*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) - 24*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh
[ArcSinh[c*x]/2]) + (-6*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 + 12*(Pi + (2*I)*ArcSinh
[c*x])*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log
[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2]))/(3*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sq
rt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (((2*I)/3)*b^2*f^2*Sqrt[I*((-I)*d + c*d*x)]
*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcSinh
[c*x] + (6 + 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*Log
[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi
*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + I*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcS
inh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*
Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24
*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2] + 24*PolyLog
[2, I/E^ArcSinh[c*x]]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f
+ c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (b^2*f^2*Sqrt[I*((-I)*d + c*d
*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(96*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2
] + I*Sinh[ArcSinh[c*x]/2]) + Sinh[ArcSinh[c*x]/2]*(24*Pi*ArcSinh[c*x] + 48*c*x*ArcSinh[c*x] + (24 - 24*I)*Arc
Sinh[c*x]^2 - (10*I)*ArcSinh[c*x]^3 + (3*I)*Sqrt[1 + c^2*x^2]*(c*x + (8*I)*(2 + ArcSinh[c*x]^2)) - (3*I)*ArcSi
nh[c*x]*Cosh[2*ArcSinh[c*x]] - 48*Pi*Log[1 - I/E^ArcSinh[c*x]] - (96*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]]
- 96*Pi*Log[1 + E^ArcSinh[c*x]] + 96*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 48*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4
]] + (3*I)*ArcSinh[c*x]^2*Sinh[2*ArcSinh[c*x]]) + Cosh[ArcSinh[c*x]/2]*(3*Sqrt[1 + c^2*x^2]*(c*x + (8*I)*(2 +
ArcSinh[c*x]^2)) - 3*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - I*(24*Pi*ArcSinh[c*x] + 48*c*x*ArcSinh[c*x] - (24 + 2
4*I)*ArcSinh[c*x]^2 - (10*I)*ArcSinh[c*x]^3 - 48*Pi*Log[1 - I/E^ArcSinh[c*x]] - (96*I)*ArcSinh[c*x]*Log[1 - I/
E^ArcSinh[c*x]] - 96*Pi*Log[1 + E^ArcSinh[c*x]] + 96*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 48*Pi*Log[Sin[(Pi + (2*I)*
ArcSinh[c*x])/4]] + (3*I)*ArcSinh[c*x]^2*Sinh[2*ArcSinh[c*x]]))))/(12*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f
*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*S
qrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-(Sinh[ArcSinh[c*x]/2]*((-16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[
c*x] + Cosh[2*ArcSinh[c*x]] + 2*((8*I)*c*x + (8*I)*ArcSinh[c*x] + 5*ArcSinh[c*x]^2 + (16*I)*ArcTan[Tanh[ArcSin
h[c*x]/2]] + 8*Log[Sqrt[1 + c^2*x^2]] - ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))) + Cosh[ArcSinh[c*x]/2]*(16*Sqrt[1
+ c^2*x^2]*ArcSinh[c*x] + I*(Cosh[2*ArcSinh[c*x]] + 2*((8*I)*c*x - (8*I)*ArcSinh[c*x] + 5*ArcSinh[c*x]^2 + (1
6*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 8*Log[Sqrt[1 + c^2*x^2]] - ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])))))/(4*c*d^2
*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]))

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Maple [F]  time = 0.274, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ( f-icfx \right ) ^{{\frac{5}{2}}} \left ( d+icdx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2),x)

[Out]

int((f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} c^{2} f^{2} x^{2} + 2 i \, b^{2} c f^{2} x - b^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} +{\left (2 \, a b c^{2} f^{2} x^{2} + 4 i \, a b c f^{2} x - 2 \, a b f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a^{2} c^{2} f^{2} x^{2} + 2 i \, a^{2} c f^{2} x - a^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c^{2} d^{2} x^{2} - 2 i \, c d^{2} x - d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((b^2*c^2*f^2*x^2 + 2*I*b^2*c*f^2*x - b^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^
2*x^2 + 1))^2 + (2*a*b*c^2*f^2*x^2 + 4*I*a*b*c*f^2*x - 2*a*b*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x
+ sqrt(c^2*x^2 + 1)) + (a^2*c^2*f^2*x^2 + 2*I*a^2*c*f^2*x - a^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))/(c
^2*d^2*x^2 - 2*I*c*d^2*x - d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f-I*c*f*x)**(5/2)*(a+b*asinh(c*x))**2/(d+I*c*d*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out