3.2.80 \(\int \frac {(d-c^2 d x^2)^3 (a+b \text {ArcSin}(c x))^2}{x^2} \, dx\) [180]

Optimal. Leaf size=329 \[ \frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))-\frac {16}{5} c^2 d^3 x (a+b \text {ArcSin}(c x))^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{x}-4 b c d^3 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )+2 i b^2 c d^3 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d^3 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right ) \]

[Out]

122/25*b^2*c^2*d^3*x-14/75*b^2*c^4*d^3*x^3+2/125*b^2*c^6*d^3*x^5-2/5*b*c*d^3*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*
x))-2/25*b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))-16/5*c^2*d^3*x*(a+b*arcsin(c*x))^2-8/5*c^2*d^3*x*(-c^2*x
^2+1)*(a+b*arcsin(c*x))^2-6/5*c^2*d^3*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-d^3*(-c^2*x^2+1)^3*(a+b*arcsin(c*x)
)^2/x-4*b*c*d^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*d^3*polylog(2,-I*c*x-(-c^2*x^2+1
)^(1/2))-2*I*b^2*c*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-22/5*b*c*d^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4785, 4743, 4715, 4767, 8, 200, 4787, 4783, 4803, 4268, 2317, 2438} \begin {gather*} -\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{x}-\frac {16}{5} c^2 d^3 x (a+b \text {ArcSin}(c x))^2-4 b c d^3 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+2 i b^2 c d^3 \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d^3 \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^2,x]

[Out]

(122*b^2*c^2*d^3*x)/25 - (14*b^2*c^4*d^3*x^3)/75 + (2*b^2*c^6*d^3*x^5)/125 - (22*b*c*d^3*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(5/2)*(
a + b*ArcSin[c*x]))/25 - (16*c^2*d^3*x*(a + b*ArcSin[c*x])^2)/5 - (8*c^2*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x
])^2)/5 - (6*c^2*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/
x - 4*b*c*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^3*PolyLog[2, -E^(I*ArcSin[c*x])]
- (2*I)*b^2*c*d^3*PolyLog[2, E^(I*ArcSin[c*x])]

Rule 8

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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 4268

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

Rule 4715

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

Rule 4743

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

Rule 4767

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

Rule 4783

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

Rule 4785

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

Rule 4787

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

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (6 c^2 d\right ) \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {1}{5} \left (24 c^2 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx+\frac {1}{5} \left (12 b c^3 d^3\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2}{3} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{5} \left (16 c^2 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac {1}{5} \left (16 b c^3 d^3\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {16}{15} b^2 c^2 d^3 x+\frac {22}{45} b^2 c^4 d^3 x^3-\frac {2}{25} b^2 c^6 d^3 x^5+2 b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{15} \left (16 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^3\right ) \int 1 \, dx+\frac {1}{5} \left (32 b c^3 d^3\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {38}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{5} \left (32 b^2 c^2 d^3\right ) \int 1 \, dx\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (2 i b^2 c d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.83, size = 483, normalized size = 1.47 \begin {gather*} \frac {1}{720} d^3 \left (-\frac {720 a^2}{x}-2160 a^2 c^2 x+3460 b^2 c^2 x+720 a^2 c^4 x^3-144 a^2 c^6 x^5-\frac {17568}{5} a b c \sqrt {1-c^2 x^2}+\frac {2016}{5} a b c^3 x^2 \sqrt {1-c^2 x^2}-\frac {288}{5} a b c^5 x^4 \sqrt {1-c^2 x^2}-\frac {1440 a b \text {ArcSin}(c x)}{x}-4320 a b c^2 x \text {ArcSin}(c x)+1440 a b c^4 x^3 \text {ArcSin}(c x)-288 a b c^6 x^5 \text {ArcSin}(c x)-3420 b^2 c \sqrt {1-c^2 x^2} \text {ArcSin}(c x)-\frac {720 b^2 \text {ArcSin}(c x)^2}{x}-1890 b^2 c^2 x \text {ArcSin}(c x)^2-1440 a b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )+80 b^2 c^2 x \cos (2 \text {ArcSin}(c x))-360 b^2 c^2 x \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))-90 b^2 c \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x))-\frac {18}{5} b^2 c \text {ArcSin}(c x) \cos (5 \text {ArcSin}(c x))+1440 b^2 c \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-1440 b^2 c \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+1440 i b^2 c \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-1440 i b^2 c \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )-10 b^2 c \sin (3 \text {ArcSin}(c x))+45 b^2 c \text {ArcSin}(c x)^2 \sin (3 \text {ArcSin}(c x))+\frac {18}{25} b^2 c \sin (5 \text {ArcSin}(c x))-9 b^2 c \text {ArcSin}(c x)^2 \sin (5 \text {ArcSin}(c x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^2,x]

[Out]

(d^3*((-720*a^2)/x - 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 - 144*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1
- c^2*x^2])/5 + (2016*a*b*c^3*x^2*Sqrt[1 - c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 - c^2*x^2])/5 - (1440*a*b*Ar
cSin[c*x])/x - 4320*a*b*c^2*x*ArcSin[c*x] + 1440*a*b*c^4*x^3*ArcSin[c*x] - 288*a*b*c^6*x^5*ArcSin[c*x] - 3420*
b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - (720*b^2*ArcSin[c*x]^2)/x - 1890*b^2*c^2*x*ArcSin[c*x]^2 - 1440*a*b*c*Ar
cTanh[Sqrt[1 - c^2*x^2]] + 80*b^2*c^2*x*Cos[2*ArcSin[c*x]] - 360*b^2*c^2*x*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] -
90*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (18*b^2*c*ArcSin[c*x]*Cos[5*ArcSin[c*x]])/5 + 1440*b^2*c*ArcSin[c*x]
*Log[1 - E^(I*ArcSin[c*x])] - 1440*b^2*c*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (1440*I)*b^2*c*PolyLog[2, -E
^(I*ArcSin[c*x])] - (1440*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])] - 10*b^2*c*Sin[3*ArcSin[c*x]] + 45*b^2*c*ArcS
in[c*x]^2*Sin[3*ArcSin[c*x]] + (18*b^2*c*Sin[5*ArcSin[c*x]])/25 - 9*b^2*c*ArcSin[c*x]^2*Sin[5*ArcSin[c*x]]))/7
20

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Maple [A]
time = 0.47, size = 464, normalized size = 1.41

method	result	size
derivativedivides	\(c \left (-d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-\frac {19 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{4}-\frac {19 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x}{8}+\frac {19 d^{3} b^{2} c x}{4}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{3} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{3} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} \sin \left (5 \arcsin \left (c x \right )\right )}{80}+\frac {d^{3} b^{2} \sin \left (5 \arcsin \left (c x \right )\right )}{1000}-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}-\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{16}+\frac {d^{3} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{24}-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(464\)
default	\(c \left (-d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-\frac {19 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{4}-\frac {19 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x}{8}+\frac {19 d^{3} b^{2} c x}{4}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{3} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{3} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} \sin \left (5 \arcsin \left (c x \right )\right )}{80}+\frac {d^{3} b^{2} \sin \left (5 \arcsin \left (c x \right )\right )}{1000}-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}-\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{16}+\frac {d^{3} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{24}-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(464\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^2,x,method=_RETURNVERBOSE)

[Out]

c*(-d^3*a^2*(1/5*c^5*x^5-c^3*x^3+3*c*x+1/c/x)-19/4*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-19/8*d^3*b^2*arcsin(
c*x)^2*c*x+19/4*d^3*b^2*c*x-d^3*b^2/c/x*arcsin(c*x)^2+2*d^3*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*d
^3*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*d^3*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*I*d^3*b^2*
polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-1/200*d^3*b^2*arcsin(c*x)*cos(5*arcsin(c*x))-1/80*d^3*b^2*arcsin(c*x)^2*s
in(5*arcsin(c*x))+1/1000*d^3*b^2*sin(5*arcsin(c*x))-1/8*d^3*b^2*arcsin(c*x)*cos(3*arcsin(c*x))-3/16*d^3*b^2*ar
csin(c*x)^2*sin(3*arcsin(c*x))+1/24*d^3*b^2*sin(3*arcsin(c*x))-2*d^3*a*b*(1/5*arcsin(c*x)*c^5*x^5-c^3*x^3*arcs
in(c*x)+3*c*x*arcsin(c*x)+1/c/x*arcsin(c*x)+1/25*c^4*x^4*(-c^2*x^2+1)^(1/2)-7/25*c^2*x^2*(-c^2*x^2+1)^(1/2)+61
/25*(-c^2*x^2+1)^(1/2)+arctanh(1/(-c^2*x^2+1)^(1/2))))

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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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^2,x, algorithm="maxima")

[Out]

-1/5*a^2*c^6*d^3*x^5 - 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4
+ 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^6*d^3 + a^2*c^4*d^3*x^3 + 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)
*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*c^4*d^3 - 3*b^2*c^2*d^3*x*arcsin(c*x)^2 + 6*b^2*c^2*d^3*(x - sqrt(-c
^2*x^2 + 1)*arcsin(c*x)/c) - 3*a^2*c^2*d^3*x - 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d^3 - 2*(c*log(2
*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*d^3 - a^2*d^3/x - 1/5*((b^2*c^6*d^3*x^6 - 5*b^2*c^
4*d^3*x^4 + 5*b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 5*x*integrate(2/5*(b^2*c^7*d^3*x^6 - 5*b
^2*c^5*d^3*x^4 + 5*b^2*c*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^3
- x), x))/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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^2,x, algorithm="fricas")

[Out]

integral(-(a^2*c^6*d^3*x^6 - 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 - a^2*d^3 + (b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^
3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*arcsin(c*x)^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^
2 - a*b*d^3)*arcsin(c*x))/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \left (- \frac {a^{2}}{x^{2}}\right )\, dx + \int \left (- 3 a^{2} c^{4} x^{2}\right )\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\right )\, dx + \int 6 a b c^{2} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b^{2} c^{4} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{4} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{2} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{4} \operatorname {asin}{\left (c x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*asin(c*x))**2/x**2,x)

[Out]

-d**3*(Integral(3*a**2*c**2, x) + Integral(-a**2/x**2, x) + Integral(-3*a**2*c**4*x**2, x) + Integral(a**2*c**
6*x**4, x) + Integral(3*b**2*c**2*asin(c*x)**2, x) + Integral(-b**2*asin(c*x)**2/x**2, x) + Integral(6*a*b*c**
2*asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**2, x) + Integral(-3*b**2*c**4*x**2*asin(c*x)**2, x) + Integral(
b**2*c**6*x**4*asin(c*x)**2, x) + Integral(-6*a*b*c**4*x**2*asin(c*x), x) + Integral(2*a*b*c**6*x**4*asin(c*x)
, x))

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^3)/x^2,x)

[Out]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^3)/x^2, x)

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