3.196 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=230 \[ \frac{i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac{i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac{b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac{b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{c d^2}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c d^2} \]

[Out]

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

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Rubi [A]  time = 0.236044, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4655, 4657, 4181, 2531, 2282, 6589, 4677, 206} \[ \frac{i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac{i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac{b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac{b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{c d^2}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac{i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*
ArcSin[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p
]), Int[x*(1 - c^2*x^2)^(p + 1/2)*(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] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 4657

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 4181

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 2531

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 2282

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 6589

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]

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 206

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

Rubi steps

\begin{align*} \int \frac{\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 d^2 \left (1-c^2 x^2\right )}-\frac{(b c) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac{b^2 \int \frac{1}{1-c^2 x^2} \, dx}{d^2}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{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 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c d^2}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c d^2}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c d^2}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{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 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c 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 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 d^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c d^2}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c d^2}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{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 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c 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 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 d^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c d^2}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{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 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c 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 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 d^2}-\frac{b^2 \text{Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac{b^2 \text{Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{c d^2}\\ \end{align*}

Mathematica [A]  time = 2.60353, size = 359, normalized size = 1.56 \[ \frac{\frac{2 a b \left (2 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-2 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+\frac{2 \left (c^2 x^2+\sqrt{1-c^2 x^2}+\sin ^{-1}(c x) \left (\left (c^2 x^2-1\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+\left (1-c^2 x^2\right ) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-c x\right )-1\right )}{c^2 x^2-1}\right )}{c}+\frac{4 b^2 \left (i \sin ^{-1}(c x) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-i \sin ^{-1}(c x) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )+\text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )+\frac{c x \sin ^{-1}(c x)^2}{2-2 c^2 x^2}-\frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\tanh ^{-1}(c x)-i \sin ^{-1}(c x)^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )\right )}{c}-\frac{2 a^2 x}{c^2 x^2-1}-\frac{a^2 \log (1-c x)}{c}+\frac{a^2 \log (c x+1)}{c}}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a^2*x)/(-1 + c^2*x^2) - (a^2*Log[1 - c*x])/c + (a^2*Log[1 + c*x])/c + (2*a*b*((2*(-1 + c^2*x^2 + Sqrt[1 -
c^2*x^2] + ArcSin[c*x]*(-(c*x) + (-1 + c^2*x^2)*Log[1 - I*E^(I*ArcSin[c*x])] + (1 - c^2*x^2)*Log[1 + I*E^(I*A
rcSin[c*x])])))/(-1 + c^2*x^2) + (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x
])]))/c + (4*b^2*(-(ArcSin[c*x]/Sqrt[1 - c^2*x^2]) + (c*x*ArcSin[c*x]^2)/(2 - 2*c^2*x^2) - I*ArcSin[c*x]^2*Arc
Tan[E^(I*ArcSin[c*x])] + ArcTanh[c*x] + I*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - I*ArcSin[c*x]*PolyL
og[2, I*E^(I*ArcSin[c*x])] - PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + PolyLog[3, I*E^(I*ArcSin[c*x])]))/c)/(4*d^2)

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Maple [B]  time = 0.15, size = 593, normalized size = 2.6 \begin{align*} -{\frac{{a}^{2}}{4\,c{d}^{2} \left ( cx-1 \right ) }}-{\frac{{a}^{2}\ln \left ( cx-1 \right ) }{4\,c{d}^{2}}}-{\frac{{a}^{2}}{4\,c{d}^{2} \left ( cx+1 \right ) }}+{\frac{{a}^{2}\ln \left ( cx+1 \right ) }{4\,c{d}^{2}}}-{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}\arcsin \left ( cx \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{2\,c{d}^{2}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{iab}{c{d}^{2}}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{{b}^{2}}{c{d}^{2}}{\it polylog} \left ( 3,-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{2\,c{d}^{2}}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{2\,i{b}^{2}}{c{d}^{2}}\arctan \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{b}^{2}}{c{d}^{2}}{\it polylog} \left ( 3,i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{iab}{c{d}^{2}}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{ab\arcsin \left ( cx \right ) x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{ab}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ab\arcsin \left ( cx \right ) }{c{d}^{2}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{ab\arcsin \left ( cx \right ) }{c{d}^{2}}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{i{b}^{2}\arcsin \left ( cx \right ) }{c{d}^{2}}{\it polylog} \left ( 2,-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{i{b}^{2}\arcsin \left ( cx \right ) }{c{d}^{2}}{\it polylog} \left ( 2,i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c*a^2/d^2/(c*x-1)-1/4/c*a^2/d^2*ln(c*x-1)-1/4/c*a^2/d^2/(c*x+1)+1/4/c*a^2/d^2*ln(c*x+1)-1/2*b^2/d^2/(c^2*
x^2-1)*arcsin(c*x)^2*x+1/c*b^2/d^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/2/c*b^2/d^2*arcsin(c*x)^2*ln(1
+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I/c*a*b/d^2*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-b^2*polylog(3,-I*(I*c*x+(-c^2
*x^2+1)^(1/2)))/c/d^2+1/2/c*b^2/d^2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*I/c*b^2/d^2*arctan(I*c*
x+(-c^2*x^2+1)^(1/2))+b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d^2-I/c*a*b/d^2*dilog(1-I*(I*c*x+(-c^2*x^2
+1)^(1/2)))-a*b/d^2/(c^2*x^2-1)*arcsin(c*x)*x+1/c*a*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-1/c*a*b/d^2*arcsin(c*
x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/c*a*b/d^2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I/c*b^2/d^2*a
rcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-I/c*b^2/d^2*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1
/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a^{2}{\left (\frac{2 \, x}{c^{2} d^{2} x^{2} - d^{2}} - \frac{\log \left (c x + 1\right )}{c d^{2}} + \frac{\log \left (c x - 1\right )}{c d^{2}}\right )} - \frac{2 \, b^{2} c x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \,{\left (c^{3} d^{2} x^{2} - c d^{2}\right )} \int \frac{4 \, a b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (2 \, b^{2} c x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x}}{4 \,{\left (c^{3} d^{2} x^{2} - c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c*d^2)) - 1/4*(2*b^2*c*x*arctan2(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*log(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^3*d^2*x^2 - c*d^2)*
integrate(1/2*(4*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (2*b^2*c*x*arctan2(c*x, sqrt(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^4*d^2*x^4 - 2*c^2*d
^2*x^2 + d^2), x))/(c^3*d^2*x^2 - c*d^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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