3.2.70 \(\int \frac {(d-c^2 d x^2)^2 (a+b \text {ArcSin}(c x))^2}{x} \, dx\) [170]

Optimal. Leaf size=271 \[ \frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {11}{32} d^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2-\frac {i d^2 (a+b \text {ArcSin}(c x))^3}{3 b}+d^2 (a+b \text {ArcSin}(c x))^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-i b d^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{2} b^2 d^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right ) \]

[Out]

13/32*b^2*c^2*d^2*x^2-1/32*b^2*c^4*d^2*x^4-1/8*b*c*d^2*x*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-11/32*d^2*(a+b*a
rcsin(c*x))^2+1/2*d^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+1/4*d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-1/3*I*d^2*(a
+b*arcsin(c*x))^3/b+d^2*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-I*b*d^2*(a+b*arcsin(c*x))*polyl
og(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*b^2*d^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-11/16*b*c*d^2*x*(a+b*ar
csin(c*x))*(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4787, 4721, 3798, 2221, 2611, 2320, 6724, 4741, 4737, 30, 4743, 14} \begin {gather*} -\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-i b d^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {i d^2 (a+b \text {ArcSin}(c x))^3}{3 b}-\frac {11}{32} d^2 (a+b \text {ArcSin}(c x))^2+d^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2+\frac {1}{2} b^2 d^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{32} b^2 c^4 d^2 x^4+\frac {13}{32} b^2 c^2 d^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13*b^2*c^2*d^2*x^2)/32 - (b^2*c^4*d^2*x^4)/32 - (11*b*c*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 - (b*
c*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/8 - (11*d^2*(a + b*ArcSin[c*x])^2)/32 + (d^2*(1 - c^2*x^2)*(a
+ b*ArcSin[c*x])^2)/2 + (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 - ((I/3)*d^2*(a + b*ArcSin[c*x])^3)/b +
d^2*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*Ar
cSin[c*x])] + (b^2*d^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2221

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4737

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

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[x*(a + b*ArcSin[c*x])^(
n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && 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 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 6724

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

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{8} \left (3 b c d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (b c d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{16} \left (3 b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^2\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^2\right ) \int x \, dx\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\left (2 i d^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (2 b d^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^2 d^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \left (b^2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} b^2 d^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 371, normalized size = 1.37 \begin {gather*} \frac {1}{768} d^2 \left (-32 i b^2 \pi ^3-768 a^2 c^2 x^2+192 a^2 c^4 x^4-624 a b c x \sqrt {1-c^2 x^2}+96 a b c^3 x^3 \sqrt {1-c^2 x^2}-1536 a b c^2 x^2 \text {ArcSin}(c x)+384 a b c^4 x^4 \text {ArcSin}(c x)-768 i a b \text {ArcSin}(c x)^2+256 i b^2 \text {ArcSin}(c x)^3+1248 a b \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-144 b^2 \cos (2 \text {ArcSin}(c x))+288 b^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))-3 b^2 \cos (4 \text {ArcSin}(c x))+24 b^2 \text {ArcSin}(c x)^2 \cos (4 \text {ArcSin}(c x))+768 b^2 \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+1536 a b \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+768 a^2 \log (c x)+768 i b^2 \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )-768 i a b \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )+384 b^2 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )-288 b^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))-12 b^2 \text {ArcSin}(c x) \sin (4 \text {ArcSin}(c x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*((-32*I)*b^2*Pi^3 - 768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 - 624*a*b*c*x*Sqrt[1 - c^2*x^2] + 96*a*b*c^3*x^3*Sq
rt[1 - c^2*x^2] - 1536*a*b*c^2*x^2*ArcSin[c*x] + 384*a*b*c^4*x^4*ArcSin[c*x] - (768*I)*a*b*ArcSin[c*x]^2 + (25
6*I)*b^2*ArcSin[c*x]^3 + 1248*a*b*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] - 144*b^2*Cos[2*ArcSin[c*x]] + 288*b^
2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 3*b^2*Cos[4*ArcSin[c*x]] + 24*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 768*
b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 1536*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 768*
a^2*Log[c*x] + (768*I)*b^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (768*I)*a*b*PolyLog[2, E^((2*I)*Ar
cSin[c*x])] + 384*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 288*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 12*b^2*Arc
Sin[c*x]*Sin[4*ArcSin[c*x]]))/768

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (277 ) = 554\).
time = 0.33, size = 560, normalized size = 2.07

method	result	size
derivativedivides	\(d^{2} a^{2} \ln \left (c x \right )-\frac {d^{2} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {3 d^{2} b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{16}+2 d^{2} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{2} a^{2} c^{4} x^{4}}{4}-d^{2} a^{2} c^{2} x^{2}+\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 d^{2} b^{2} \arcsin \left (c x \right )^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {3 d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {d^{2} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {3 d^{2} a b \sin \left (2 \arcsin \left (c x \right )\right )}{8}+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {i d^{2} b^{2} \arcsin \left (c x \right )^{3}}{3}+\frac {d^{2} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {3 d^{2} a b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{2} a b \arcsin \left (c x \right )^{2}-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\)	\(560\)
default	\(d^{2} a^{2} \ln \left (c x \right )-\frac {d^{2} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {3 d^{2} b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{16}+2 d^{2} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{2} a^{2} c^{4} x^{4}}{4}-d^{2} a^{2} c^{2} x^{2}+\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 d^{2} b^{2} \arcsin \left (c x \right )^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {3 d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {d^{2} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {3 d^{2} a b \sin \left (2 \arcsin \left (c x \right )\right )}{8}+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {i d^{2} b^{2} \arcsin \left (c x \right )^{3}}{3}+\frac {d^{2} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {3 d^{2} a b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{2} a b \arcsin \left (c x \right )^{2}-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\)	\(560\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*a^2*ln(c*x)+2*d^2*b^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))+2*d^2*b^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-1
/256*d^2*b^2*cos(4*arcsin(c*x))-3/16*d^2*b^2*cos(2*arcsin(c*x))-2*I*d^2*b^2*arcsin(c*x)*polylog(2,I*c*x+(-c^2*
x^2+1)^(1/2))-2*I*d^2*b^2*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*d^2*a*b*arcsin(c*x)*ln(1-I*c*x-(-
c^2*x^2+1)^(1/2))+1/4*d^2*a^2*c^4*x^4-d^2*a^2*c^2*x^2+d^2*b^2*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+d^2
*b^2*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/32*d^2*b^2*arcsin(c*x)^2*cos(4*arcsin(c*x))-1/64*d^2*b^2*a
rcsin(c*x)*sin(4*arcsin(c*x))+3/8*d^2*b^2*arcsin(c*x)^2*cos(2*arcsin(c*x))-3/8*d^2*b^2*arcsin(c*x)*sin(2*arcsi
n(c*x))-1/3*I*d^2*b^2*arcsin(c*x)^3-1/64*d^2*a*b*sin(4*arcsin(c*x))-3/8*d^2*a*b*sin(2*arcsin(c*x))+2*d^2*a*b*a
rcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/16*d^2*a*b*arcsin(c*x)*cos(4*arcsin(c*x))+3/4*d^2*a*b*arcsin(c*x)*
cos(2*arcsin(c*x))-I*d^2*a*b*arcsin(c*x)^2-2*I*d^2*a*b*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-2*I*d^2*a*b*polylog
(2,-I*c*x-(-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)^2*(a+b*arcsin(c*x))^2/x,x, algorithm="maxima")

[Out]

1/4*a^2*c^4*d^2*x^4 - a^2*c^2*d^2*x^2 + a^2*d^2*log(x) + integrate(((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2
*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arctan2
(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/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)^2*(a+b*arcsin(c*x))^2/x,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [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)^2*(a+b*arcsin(c*x))^2/x,x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 - d)^2*(b*arcsin(c*x) + a)^2/x, x)

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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 )}^2}{x} \,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)^2)/x,x)

[Out]

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

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