\(\int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) [596]

Optimal result

Integrand size = 37, antiderivative size = 544 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]

[Out]

-2*I*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+2*d^2*x*(c^2*x^2+1)*(a+b*arcsi
nh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+2*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2
)/(f-I*c*f*x)^(3/2)-1/3*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^3/b/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*I
*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2
)-4*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)
^(3/2)+4*b^2*d^2*(c^2*x^2+1)^(3/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)
-4*b^2*d^2*(c^2*x^2+1)^(3/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-2*b^2*
d^2*(c^2*x^2+1)^(3/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {5796, 5844, 5838, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 5789, 4265, 5783} \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {8 i b d^2 \left (c^2 x^2+1\right )^{3/2} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 i d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (c^2 x^2+1\right )^{3/2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 b^2 d^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b^2 d^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]

[In]

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

[Out]

((-2*I)*d^2*(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)) + (2*d^2*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)) + (2*d^2*(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)) - (d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^
3)/(3*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Arc
Tan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b*d^2*(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)) + (4*b^2*d^2*(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)) - (4*b^2*d^2*(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)) - (2*b^2*d^2*(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 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 3799

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 4265

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 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 5787

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/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[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 5789

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 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 5797

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 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 5838

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 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 )^{3/2} \int \frac {(d+i c d x)^2 (a+b \text {arcsinh}(c x))^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 {2 i \left (i d^2-c d^2 x\right ) (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {d^2 (a+b \text {arcsinh}(c x))^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 (2 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\left (i d^2-c d^2 x\right ) (a+b \text {arcsinh}(c x))^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 (d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (\frac {i d^2 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {c d^2 x (a+b \text {arcsinh}(c x))^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 {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^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 (2 i c d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^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 {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (4 i b d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b c d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (4 i b d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (8 b d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\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)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\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)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\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)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\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)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 6.69 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {\frac {6 a^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{i+c x}-3 a^2 \sqrt {d} \sqrt {f} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-\frac {b^2 (-i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-18 \pi \text {arcsinh}(c x)-(6-6 i) \text {arcsinh}(c x)^2+i \text {arcsinh}(c x)^3-12 (\pi -2 i \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+24 \pi \log \left (1+e^{\text {arcsinh}(c x)}\right )+12 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )-24 \pi \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-24 i \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\frac {12 i \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^2}+\frac {3 a b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 \text {arcsinh}(c x) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{3 c f^2} \]

[In]

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

[Out]

((6*a^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(I + c*x) - 3*a^2*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*S
qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - (b^2*(-I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-18*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*P
i*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (
24*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I
*Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (3*a*b*Sqrt[d
+ I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c
*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2
])*(I*Cosh[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*c*f^2)

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {i c d x +d}}{\left (-i c f x +f \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2),x)

[Out]

int((a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2),x)

Fricas [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int \frac {\sqrt {i d \left (c x - i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*asinh(c*x))**2*(d+I*c*d*x)**(1/2)/(f-I*c*f*x)**(3/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2),x, algorithm="maxima")

[Out]

a^2*(-2*I*sqrt(c^2*d*f*x^2 + d*f)/(-I*c^2*f^2*x + c*f^2) - d*arcsinh(c*x)/(c*f^2*sqrt(d/f))) + integrate(sqrt(
I*c*d*x + d)*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(-I*c*f*x + f)^(3/2) + 2*sqrt(I*c*d*x + d)*a*b*log(c*x + sqrt(
c^2*x^2 + 1))/(-I*c*f*x + f)^(3/2), x)

Giac [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

[In]

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(1/2))/(f - c*f*x*1i)^(3/2),x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(1/2))/(f - c*f*x*1i)^(3/2), x)