\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx\) [605]

Optimal result

Integrand size = 37, antiderivative size = 386 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=-\frac {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

[Out]

-1/3*b^2*x*(c^2*x^2+1)^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+1/3*b*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c/(d+I
*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+1/3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+2/3
*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+2/3*(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)-4/3*b*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^
(1/2))^2)/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)-2/3*b^2*(c^2*x^2+1)^(5/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2
)/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {5796, 5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b \left (c^2 x^2+1\right )^{5/2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 x \left (c^2 x^2+1\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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 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 5788

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(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 {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 \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 {\left (8 b \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 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 {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 \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 {4 b \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 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 {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 \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 {4 b \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {b^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 \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 {4 b \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.66 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\frac {4 a^2 c x \left (3+2 c^2 x^2\right )-b^2 \left (c x-6 c x \text {arcsinh}(c x)^2+4 i \pi \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x))+2 \text {arcsinh}(c x)^2 \cosh (3 \text {arcsinh}(c x))-2 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x)) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+2 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-8 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (1+e^{\text {arcsinh}(c x)}\right )-2 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+8 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 i \pi \cosh (3 \text {arcsinh}(c x)) \log \left (\sin \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+2 \sqrt {1+c^2 x^2} \left ((-3 i \pi +6 \text {arcsinh}(c x)) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+i \left (2 i \text {arcsinh}(c x)+6 \pi \text {arcsinh}(c x)-3 i \text {arcsinh}(c x)^2+3 (\pi -2 i \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-12 \pi \log \left (1+e^{\text {arcsinh}(c x)}\right )-3 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+12 \pi \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+3 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )\right )\right )-16 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-16 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+\sinh (3 \text {arcsinh}(c x))-2 \text {arcsinh}(c x)^2 \sinh (3 \text {arcsinh}(c x))\right )+2 a b \left (\sqrt {1+c^2 x^2} \left (2-3 \log \left (1+c^2 x^2\right )\right )-\cosh (3 \text {arcsinh}(c x)) \log \left (1+c^2 x^2\right )+2 \text {arcsinh}(c x) (3 c x+\sinh (3 \text {arcsinh}(c x)))\right )}{12 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (c+c^3 x^2\right )} \]

[In]

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

[Out]

(4*a^2*c*x*(3 + 2*c^2*x^2) - b^2*(c*x - 6*c*x*ArcSinh[c*x]^2 + (4*I)*Pi*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] + 2*
ArcSinh[c*x]^2*Cosh[3*ArcSinh[c*x]] - (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + 4*ArcSinh[c*x]
*Cosh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 + I/E^ArcSinh[c*x]] + 4*
ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]]*Log[1 + I/E^ArcSinh[c*x]] - (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 + E^ArcSinh[
c*x]] - (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*L
og[Cosh[ArcSinh[c*x]/2]] + (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + 2*Sqrt[1 + c^
2*x^2]*(((-3*I)*Pi + 6*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + I*((2*I)*ArcSinh[c*x] + 6*Pi*ArcSinh[c*x] - (
3*I)*ArcSinh[c*x]^2 + 3*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - 12*Pi*Log[1 + E^ArcSinh[c*x]] -
3*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 12*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 3*Pi*Log[Sin[(Pi + (2*I)*ArcSi
nh[c*x])/4]])) - 16*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - 16*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/
E^ArcSinh[c*x]] + Sinh[3*ArcSinh[c*x]] - 2*ArcSinh[c*x]^2*Sinh[3*ArcSinh[c*x]]) + 2*a*b*(Sqrt[1 + c^2*x^2]*(2
- 3*Log[1 + c^2*x^2]) - Cosh[3*ArcSinh[c*x]]*Log[1 + c^2*x^2] + 2*ArcSinh[c*x]*(3*c*x + Sinh[3*ArcSinh[c*x]]))
)/(12*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(c + c^3*x^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

1/3*((2*b^2*c^2*x^3 + 3*b^2*x)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 3*(c^4*d^
3*f^3*x^4 + 2*c^2*d^3*f^3*x^2 + d^3*f^3)*integral(1/3*(3*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a^2 + 2*(3*sqrt(
I*c*d*x + d)*sqrt(-I*c*f*x + f)*a*b - (2*b^2*c^3*x^3 + 3*b^2*c*x)*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*
c*f*x + f))*log(c*x + sqrt(c^2*x^2 + 1)))/(c^6*d^3*f^3*x^6 + 3*c^4*d^3*f^3*x^4 + 3*c^2*d^3*f^3*x^2 + d^3*f^3),
x))/(c^4*d^3*f^3*x^4 + 2*c^2*d^3*f^3*x^2 + d^3*f^3)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/3*a*b*c*(1/(c^4*d^(5/2)*f^(5/2)*x^2 + c^2*d^(5/2)*f^(5/2)) - 2*log(c^2*x^2 + 1)/(c^2*d^(5/2)*f^(5/2))) + 2/3
*a*b*(x/((c^2*d*f*x^2 + d*f)^(3/2)*d*f) + 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d^2*f^2))*arcsinh(c*x) + 1/3*a^2*(x/((c
^2*d*f*x^2 + d*f)^(3/2)*d*f) + 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d^2*f^2)) + b^2*integrate(log(c*x + sqrt(c^2*x^2 +
1))^2/((I*c*d*x + d)^(5/2)*(-I*c*f*x + f)^(5/2)), x)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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