\(\int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx\) [587]

Optimal result

Integrand size = 37, antiderivative size = 790 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

[Out]

2*I*a*b*f^5*x*(c^2*x^2+1)^(5/2)/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)-2*I*b^2*f^5*(c^2*x^2+1)^3/c/(d+I*c*d*x)^(5
/2)/(f-I*c*f*x)^(5/2)+2*I*b^2*f^5*x*(c^2*x^2+1)^(5/2)*arcsinh(c*x)/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)-28/3*f^
5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)-I*f^5*(c^2*x^2+1)^3*(a+b*arcsin
h(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+5/3*f^5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))^3/b/c/(d+I*c*d*x)
^(5/2)/(f-I*c*f*x)^(5/2)-16/3*I*b^2*f^5*(c^2*x^2+1)^(5/2)*cot(1/4*Pi+1/2*I*arcsinh(c*x))/c/(d+I*c*d*x)^(5/2)/(
f-I*c*f*x)^(5/2)-28/3*I*f^5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))^2*cot(1/4*Pi+1/2*I*arcsinh(c*x))/c/(d+I*c*d*x
)^(5/2)/(f-I*c*f*x)^(5/2)+8/3*b*f^5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))*csc(1/4*Pi+1/2*I*arcsinh(c*x))^2/c/(d
+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+4/3*I*f^5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))^2*cot(1/4*Pi+1/2*I*arcsinh(c*
x))*csc(1/4*Pi+1/2*I*arcsinh(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+112/3*b*f^5*(c^2*x^2+1)^(5/2)*(a+b*
arcsinh(c*x))*ln(1+I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+112/3*b^2*f^5*(c^2*x^2+1)^
(5/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {5796, 5844, 5783, 5798, 5772, 267, 5843, 3399, 4271, 3852, 8, 4269, 3797, 2221, 2317, 2438} \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\frac {5 f^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b f^5 \left (c^2 x^2+1\right )^{5/2} \log \left (1+i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (c^2 x^2+1\right )^{5/2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i a b f^5 x \left (c^2 x^2+1\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b^2 f^5 \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (c^2 x^2+1\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (c^2 x^2+1\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

[In]

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

[Out]

((2*I)*a*b*f^5*x*(1 + c^2*x^2)^(5/2))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((2*I)*b^2*f^5*(1 + c^2*x^2)
^3)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((2*I)*b^2*f^5*x*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x])/((d + I*c
*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (28*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2
)*(f - I*c*f*x)^(5/2)) - (I*f^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(
5/2)) + (5*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (
((16*I)/3)*b^2*f^5*(1 + c^2*x^2)^(5/2)*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5
/2)) - (((28*I)/3)*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*
d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (8*b*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcSinh[c
*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((4*I)/3)*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]
)^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]]*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2
)) + (112*b*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(
f - I*c*f*x)^(5/2)) + (112*b^2*f^5*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/
2)*(f - I*c*f*x)^(5/2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 267

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 2221

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

Rule 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 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 3399

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

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 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 5772

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 5783

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

Rule 5796

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 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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 5843

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

Rule 5844

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(f-i c f x)^5 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (1+c^2 x^2\right )^{5/2} \int \left (\frac {5 f^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {i c f^5 x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {8 f^5 (a+b \text {arcsinh}(c x))^2}{(-i+c x)^2 \sqrt {1+c^2 x^2}}+\frac {12 i f^5 (a+b \text {arcsinh}(c x))^2}{(-i+c x) \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (12 i f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{(-i+c x) \sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (5 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{(-i+c x)^2 \sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i c f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (12 i f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{-i c+c \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 c f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{(-i c+c \sinh (x))^2} \, dx,x,\text {arcsinh}(c x)\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \text {arcsinh}(c x) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (6 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {12 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (24 i b f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \csc ^2\left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b^2 c f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {12 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (16 i b f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {i x}{2}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (48 i b f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{1+i e^x} \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int 1 \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {48 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (32 i b f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{1+i e^x} \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (48 b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (32 b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (48 b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {48 b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (32 b^2 f^5 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i a b f^5 x \left (1+c^2 x^2\right )^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b^2 f^5 \left (1+c^2 x^2\right )^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 f^5 x \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {5 f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {16 i b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {28 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b f^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {112 b^2 f^5 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2622\) vs. \(2(790)=1580\).

Time = 24.70 (sec) , antiderivative size = 2622, normalized size of antiderivative = 3.32 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-I)*a^2*f^2)/d^3 - (((8*I)/3)*a^2*f^2)/(d^3*(-I + c*x)^2) - (2
8*a^2*f^2)/(3*d^3*(-I + c*x))))/c + (5*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)]])/(c*d^(5/2)) + ((I/3)*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] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*
ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I
)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Cot
h[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[
Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[Arc
Sinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) - (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] - I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((-14 + (3*I)*Ar
cSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh[ArcSinh[c*x]/2]] + (14*I)*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x
]/2]*(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + 42*Log[Sqrt[1 + c^2*x^2
]])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 + (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^
2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(-14*I + 3*ArcSinh[c*x]) + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*
Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(3*c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Co
sh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((I/3)*b^2*f^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*
(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 + I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-
I + c*x) + (2*I)*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - I*Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSin
h[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + 4*PolyLog[2, I/E^ArcSinh[c*
x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcS
inh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^3*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])^2) - ((I/3)*b^2*f^2*(I
+ c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(((6*I)*c*x*ArcSinh[c*x])/
Sqrt[1 + c^2*x^2] + ((13 - 13*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (3*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (2
*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/((-I + c*x)*Sqrt[1 + c^2*x^2]) - (3*I)*(2 + ArcSinh[c*x]^2) + ((13*I)*(-2
*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Co
sh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + (4*I)*PolyLog[2, I/E^ArcSinh[c*x]]))/Sqrt[1 +
c^2*x^2] + (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[
c*x]/2])^3) - (2*(4 + 13*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Si
nh[ArcSinh[c*x]/2]))))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c
*x]/2])^2) + (2*b^2*f^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]
*(7*Pi*ArcSinh[c*x] - (7 + 7*I)*ArcSinh[c*x]^2 - I*ArcSinh[c*x]^3 + (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(1
+ I*c*x) - 14*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - 28*Pi*Log[1 + E^ArcSinh[c*x]] + 28*Pi*Log[
Cosh[ArcSinh[c*x]/2]] + 14*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (28*I)*PolyLog[2, I/E^ArcSinh[c*x]] - ((
4*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + 7*ArcSin
h[c*x]^2)*Sinh[ArcSinh[c*x]/2])/((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(3*c*d^3*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])^2) + ((I/6)*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] - I*Sinh[Ar
cSinh[c*x]/2])*(-3*Cosh[(5*ArcSinh[c*x])/2] + (3*I)*ArcSinh[c*x]*Cosh[(5*ArcSinh[c*x])/2] - Cosh[(3*ArcSinh[c*
x])/2]*(9 + (35*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2
*x^2]]) + Cosh[ArcSinh[c*x]/2]*(20 - (24*I)*ArcSinh[c*x] + 27*ArcSinh[c*x]^2 - (156*I)*ArcTan[Coth[ArcSinh[c*x
]/2]] + 78*Log[Sqrt[1 + c^2*x^2]]) + (20*I)*Sinh[ArcSinh[c*x]/2] - 24*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2] + (27*
I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2] + 156*ArcTan[Coth[ArcSinh[c*x]/2]]*Sinh[ArcSinh[c*x]/2] + (78*I)*Log[Sq
rt[1 + c^2*x^2]]*Sinh[ArcSinh[c*x]/2] + (9*I)*Sinh[(3*ArcSinh[c*x])/2] + 35*ArcSinh[c*x]*Sinh[(3*ArcSinh[c*x])
/2] + (9*I)*ArcSinh[c*x]^2*Sinh[(3*ArcSinh[c*x])/2] + 52*ArcTan[Coth[ArcSinh[c*x]/2]]*Sinh[(3*ArcSinh[c*x])/2]
+ (26*I)*Log[Sqrt[1 + c^2*x^2]]*Sinh[(3*ArcSinh[c*x])/2] - (3*I)*Sinh[(5*ArcSinh[c*x])/2] + 3*ArcSinh[c*x]*Si
nh[(5*ArcSinh[c*x])/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Si
nh[ArcSinh[c*x]/2])^4)

Maple [F]

\[\int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (i c d x +d \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integral(((-I*b^2*c^2*f^2*x^2 + 2*b^2*c*f^2*x + I*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*(I*a*b*c^2*f^2*x^2 - 2*a*b*c*f^2*x - I*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)) + (-I*a^2*c^2*f^2*x^2 + 2*a^2*c*f^2*x + I*a^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x +
f))/(c^3*d^3*x^3 - 3*I*c^2*d^3*x^2 - 3*c*d^3*x + I*d^3), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-2)]

Exception generated. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]

[In]

int(((a + b*asinh(c*x))^2*(f - c*f*x*1i)^(5/2))/(d + c*d*x*1i)^(5/2),x)

[Out]

int(((a + b*asinh(c*x))^2*(f - c*f*x*1i)^(5/2))/(d + c*d*x*1i)^(5/2), x)