3.6.94 \(\int \frac {(d+i c d x)^{5/2} (a+b \sinh ^{-1}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) [594]

Optimal. Leaf size=972 \[ \frac {8 i a b d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\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 d^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 d^4 \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-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 d^4 \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,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 d^4 \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]

[Out]

8*I*a*b*d^4*x*(c^2*x^2+1)^(3/2)/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-8*I*b^2*d^4*(c^2*x^2+1)^2/c/(d+I*c*d*x)^(3
/2)/(f-I*c*f*x)^(3/2)+1/4*b^2*d^4*x*(c^2*x^2+1)^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-1/4*b^2*d^4*(c^2*x^2+1)^
(3/2)*arcsinh(c*x)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-8*I*d^4*(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)-1/2*b*c*d^4*x^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(
3/2)+32*I*b*d^4*(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)+8*d^4*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*d^4*(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)-4*I*d^4*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/c/(d+I
*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+1/2*d^4*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/
2)-5/2*d^4*(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^2*d^4*x*(c^2*x
^2+1)^(3/2)*arcsinh(c*x)/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-16*b*d^4*(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)+16*b^2*d^4*(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)-16*b^2*d^4*(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)-8*b^2*d^4*(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]
time = 0.90, antiderivative size = 972, normalized size of antiderivative = 1.00, 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 = {5796, 5844, 5838, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 5789, 4265, 5783, 5772, 267, 5812, 5776, 327, 221} \begin {gather*} -\frac {5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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} d^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) d^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) d^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 ) d^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 ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) d^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 ) d^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 {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right ) d^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 {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right ) d^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 {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*I)*a*b*d^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*d^4*(1 + c^2*x^2)
^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b^2*d^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*d^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*d^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*d^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)*d^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*d^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*d^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)*d^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)) + (d^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*d^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*d^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*d^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*d^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*d^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*d^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 221

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]

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 327

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 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 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 5776

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 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 5812

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

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*} \int \frac {(d+i c d x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{(f-i c f x)^{3/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(d+i c d 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 d^4-c d^4 x\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {7 d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {4 i c d^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {c^2 d^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 d^4-c d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {c d^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 (d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^4 \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,\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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2143\) vs. \(2(972)=1944\).
time = 11.57, size = 2143, normalized size = 2.20 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-4*I)*a^2*d^2)/f^2 + (a^2*c*d^2*x)/(2*f^2) + (8*a^2*d^2)/(f^2*
(I + c*x))))/c - (15*a^2*d^(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*f^(3/2)) - ((4*I)*a*b*d^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*I)*ArcSinh[c*x] + I*Sqrt[1 + c^2*x^2]*ArcS
inh[c*x] + ArcSinh[c*x]^2 + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 2*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2
]))/(c*f^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*d^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]*(8*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + 4*Log[Sqrt[1 + c^2*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]])*Sinh[ArcS
inh[c*x]/2]))/(c*f^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])) - (b^2*d^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))]*(-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*Pi*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Lo
g[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])))/(3*c*f^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])^2) - (((2*I)/3)*b^2*d^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] + ((6
+ 6*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (3*I)*(2 + ArcSinh[c*x]^2) +
((6*I)*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]
] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[
c*x]]))/Sqrt[1 + c^2*x^2] - (12*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]))))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[Arc
Sinh[c*x]/2])^2) + (b^2*d^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))]*((-96*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((48 - 48*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] - ((20*I)*Arc
Sinh[c*x]^3)/Sqrt[1 + c^2*x^2] + 48*(2 + ArcSinh[c*x]^2) + (6*I)*c*x*(1 + 2*ArcSinh[c*x]^2) - ((6*I)*ArcSinh[c
*x]*Cosh[2*ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (48*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*
(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*
x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + ((96*I)*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]))))/(24*c*f^2*Sqrt[-(((-I)*d + c*d*x)*(
I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I
)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Sinh[ArcSinh[c*x]/2]*(-16*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*Cosh
[2*ArcSinh[c*x]] + 2*(8*c*x + 8*ArcSinh[c*x] + (5*I)*ArcSinh[c*x]^2 + 16*ArcTan[Tanh[ArcSinh[c*x]/2]] + (8*I)*
Log[Sqrt[1 + c^2*x^2]] - I*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])) - Cosh[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[T
anh[ArcSinh[c*x]/2]] - 8*Log[Sqrt[1 + c^2*x^2]] + ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))))/(4*c*f^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]))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (i c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (-i c f x +f \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c^2*d^3*x^3/(sqrt(c^2*d*f*x^2 + d*f)*f) - 8*I*c*d^3*x^2/(sqrt(c^2*d*f*x^2 + d*f)*f) + 17*d^3*x/(sqrt(c^2*
d*f*x^2 + d*f)*f) - 15*d^3*arcsinh(c*x)/(sqrt(d*f)*c*f) - 24*I*d^3/(sqrt(c^2*d*f*x^2 + d*f)*c*f))*a^2 + integr
ate((I*c*d*x + d)^(5/2)*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(-I*c*f*x + f)^(3/2) + 2*(I*c*d*x + d)^(5/2)*a*b*lo
g(c*x + sqrt(c^2*x^2 + 1))/(-I*c*f*x + f)^(3/2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(5/2))/(f - c*f*x*1i)^(3/2),x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(5/2))/(f - c*f*x*1i)^(3/2), x)

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