3.2.94 \(\int \frac {x^2 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^2} \, dx\) [194]

Optimal. Leaf size=233 \[ -\frac {b (a+b \text {ArcSin}(c x))}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \text {ArcSin}(c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i (a+b \text {ArcSin}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2}-\frac {i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}+\frac {i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}+\frac {b^2 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}-\frac {b^2 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2} \]

[Out]

1/2*x*(a+b*arcsin(c*x))^2/c^2/d^2/(-c^2*x^2+1)+I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c^3/d^2+
b^2*arctanh(c*x)/c^3/d^2-I*b*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2+I*b*(a+b*arcsi
n(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2+b^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2-b
^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2-b*(a+b*arcsin(c*x))/c^3/d^2/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4791, 4749, 4266, 2611, 2320, 6724, 4767, 212} \begin {gather*} \frac {i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{c^3 d^2}-\frac {i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d^2}+\frac {i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d^2}+\frac {x (a+b \text {ArcSin}(c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b (a+b \text {ArcSin}(c x))}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {b^2 \text {Li}_3\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}-\frac {b^2 \text {Li}_3\left (i e^{i \text {ArcSin}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(a + b*ArcSin[c*x]))/(c^3*d^2*Sqrt[1 - c^2*x^2])) + (x*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) +
(I*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d^2) + (b^2*ArcTanh[c*x])/(c^3*d^2) - (I*b*(a + b*Ar
cSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^2) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c
*x])])/(c^3*d^2) + (b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^2) - (b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/
(c^3*d^2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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 4266

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

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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 4791

Int[((a_.) + ArcSin[(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*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p + 1
))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[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 {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c d^2}-\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3 d^2}+\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2}+\frac {b \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d^2}-\frac {b \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d^2}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d^2}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d^2}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}-\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{c^3 d^2}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}+\frac {b^2 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^2}-\frac {b^2 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d^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. \(727\) vs. \(2(233)=466\).
time = 3.77, size = 727, normalized size = 3.12 \begin {gather*} -\frac {\frac {2 a^2 c x}{-1+c^2 x^2}+\frac {2 b^2 \text {ArcSin}(c x) \left (-2 \sqrt {1-c^2 x^2}+c x \text {ArcSin}(c x)\right )}{-1+c^2 x^2}+\frac {2 a b \left (1-2 \sqrt {1-c^2 x^2}+\cos (2 \text {ArcSin}(c x))+\text {ArcSin}(c x) \left (2 c x-\log \left (1-i e^{i \text {ArcSin}(c x)}\right )+\log \left (1+i e^{i \text {ArcSin}(c x)}\right )+\cos (2 \text {ArcSin}(c x)) \left (-\log \left (1-i e^{i \text {ArcSin}(c x)}\right )+\log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )\right )\right )}{-1+c^2 x^2}-a^2 \log (1-c x)+a^2 \log (1+c x)+4 i a b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-4 i a b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )+2 b^2 \left (\text {ArcSin}(c x)^2 \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+\pi \text {ArcSin}(c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (1-i e^{i \text {ArcSin}(c x)}\right )\right )-\text {ArcSin}(c x)^2 \log \left (1+i e^{i \text {ArcSin}(c x)}\right )-\text {ArcSin}(c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (-i+e^{i \text {ArcSin}(c x)}\right )\right )+\pi \text {ArcSin}(c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (-i+e^{i \text {ArcSin}(c x)}\right )\right )+\text {ArcSin}(c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left ((1+i)+(1-i) e^{i \text {ArcSin}(c x)}\right )\right )-\pi \text {ArcSin}(c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+\text {ArcSin}(c x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-\text {ArcSin}(c x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-\pi \text {ArcSin}(c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+2 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-2 i \text {ArcSin}(c x) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )-2 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )+2 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )\right )}{4 c^3 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*((2*a^2*c*x)/(-1 + c^2*x^2) + (2*b^2*ArcSin[c*x]*(-2*Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/(-1 + c^2*x^2)
+ (2*a*b*(1 - 2*Sqrt[1 - c^2*x^2] + Cos[2*ArcSin[c*x]] + ArcSin[c*x]*(2*c*x - Log[1 - I*E^(I*ArcSin[c*x])] +
Log[1 + I*E^(I*ArcSin[c*x])] + Cos[2*ArcSin[c*x]]*(-Log[1 - I*E^(I*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])
]))))/(-1 + c^2*x^2) - a^2*Log[1 - c*x] + a^2*Log[1 + c*x] + (4*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4
*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] + 2*b^2*(ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + Pi*ArcSin[c*x]*L
og[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x
])] - ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + Pi*ArcSin[c*x]*Log[-1/
2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*Arc
Sin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] - Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*Log[Cos[ArcSin[c*
x]/2] - Sin[ArcSin[c*x]/2]] + ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 2*Log[Cos[ArcSin[c*
x]/2] + Sin[ArcSin[c*x]/2]] - ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - Pi*ArcSin[c*x]*Log[
Sin[(Pi + 2*ArcSin[c*x])/4]] + (2*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLo
g[2, I*E^(I*ArcSin[c*x])] - 2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 2*PolyLog[3, I*E^(I*ArcSin[c*x])]))/(c^3*d^
2)

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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. 547 vs. \(2 (262 ) = 524\).
time = 0.32, size = 548, normalized size = 2.35

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{4 d^{2} \left (c x +1\right )}-\frac {a^{2} \ln \left (c x +1\right )}{4 d^{2}}-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {a^{2} \ln \left (c x -1\right )}{4 d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{3}}\)	\(548\)
default	\(\frac {-\frac {a^{2}}{4 d^{2} \left (c x +1\right )}-\frac {a^{2} \ln \left (c x +1\right )}{4 d^{2}}-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {a^{2} \ln \left (c x -1\right )}{4 d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{3}}\)	\(548\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-1/4*a^2/d^2/(c*x+1)-1/4*a^2/d^2*ln(c*x+1)-1/4*a^2/d^2/(c*x-1)+1/4*a^2/d^2*ln(c*x-1)-1/2*b^2/d^2/(c^2*x
^2-1)*arcsin(c*x)^2*c*x+b^2/d^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/2*b^2/d^2*arcsin(c*x)^2*ln(1-I*(I
*c*x+(-c^2*x^2+1)^(1/2)))-2*I*b^2/d^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-b^2/d^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)
^(1/2)))+1/2*b^2/d^2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*a*b/d^2*dilog(1-I*(I*c*x+(-c^2*x^2+1)^
(1/2)))+b^2/d^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-I*b^2/d^2*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1
)^(1/2)))-a*b/d^2/(c^2*x^2-1)*arcsin(c*x)*c*x+a*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+a*b/d^2*arcsin(c*x)*ln(1+
I*(I*c*x+(-c^2*x^2+1)^(1/2)))-a*b/d^2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*b^2/d^2*arcsin(c*x)*pol
ylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-I*a*b/d^2*dilog(1+I*(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(x^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

-1/4*a^2*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d^2)) - 1/4*(2*b^2*c*x*arct
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*lo
g(c*x + 1) - (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) + 4*(c^5*d^2*x^2 -
c^3*d^2)*integrate(-1/2*(4*a*b*c^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (2*b^2*c*x*arctan2(c*x, s
qrt(c*x + 1)*sqrt(-c*x + 1)) + (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - (
b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^
6*d^2*x^4 - 2*c^4*d^2*x^2 + c^2*d^2), x))/(c^5*d^2*x^2 - c^3*d^2)

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} x^{2}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2*x**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x**2*asin(c*x)**2/(c**4*x**4 - 2*c**2*x*
*2 + 1), x) + Integral(2*a*b*x**2*asin(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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