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

Optimal. Leaf size=79 \[ -\frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^2 c^3}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^2 c^3}-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-(x^2/(b*c*(a + b*ArcSin[c*x]))) - (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^2*c^3) + (Cos[(2*a
)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^3)

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Rubi [A]  time = 0.244308, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4719, 4635, 4406, 12, 3303, 3299, 3302} \[ -\frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^3}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^3}-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]

[Out]

-(x^2/(b*c*(a + b*ArcSin[c*x]))) - (CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(b^2*c^3) + (Cos[(2*a)/
b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(b^2*c^3)

Rule 4719

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

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \int \frac{x}{a+b \sin ^{-1}(c x)} \, dx}{b c}\\ &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 (a+b x)} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\cos \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac{\sin \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right ) \sin \left (\frac{2 a}{b}\right )}{b^2 c^3}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.156889, size = 70, normalized size = 0.89 \[ \frac{-\frac{b c^2 x^2}{a+b \sin ^{-1}(c x)}-\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{b^2 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]

[Out]

(-((b*c^2*x^2)/(a + b*ArcSin[c*x])) - CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + Cos[(2*a)/b]*SinIntegr
al[2*(a/b + ArcSin[c*x])])/(b^2*c^3)

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Maple [A]  time = 0.047, size = 136, normalized size = 1.7 \begin{align*}{\frac{1}{2\,{b}^{2}{c}^{3} \left ( a+b\arcsin \left ( cx \right ) \right ) } \left ( 2\,\arcsin \left ( cx \right ){\it Si} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) b-2\,\arcsin \left ( cx \right ){\it Ci} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) b+2\,{\it Si} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) a-2\,{\it Ci} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) a+\cos \left ( 2\,\arcsin \left ( cx \right ) \right ) b-b \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2),x)

[Out]

1/2/c^3*(2*arcsin(c*x)*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*b-2*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b
+2*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*a-2*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a+cos(2*arcsin(c*x))*b-b)/b^2/(a+
b*arcsin(c*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{a^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^2/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - a^2 + 2*(a*b*c^2*x^2 - a*b
)*arcsin(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*asin(c*x))**2/(-c**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.48108, size = 467, normalized size = 5.91 \begin{align*} -\frac{2 \, b \arcsin \left (c x\right ) \cos \left (\frac{a}{b}\right ) \operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac{2 \, b \arcsin \left (c x\right ) \cos \left (\frac{a}{b}\right )^{2} \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac{2 \, a \cos \left (\frac{a}{b}\right ) \operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac{2 \, a \cos \left (\frac{a}{b}\right )^{2} \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac{b \arcsin \left (c x\right ) \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )} b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac{a \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac{b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-2*b*arcsin(c*x)*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 2*b
*arcsin(c*x)*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2*a*cos(a/b)*c
os_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 2*a*cos(a/b)^2*sin_integral(2*
a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*arcsin(c*x)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^
3*c^3*arcsin(c*x) + a*b^2*c^3) - (c^2*x^2 - 1)*b/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - a*sin_integral(2*a/b + 2*
arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b/(b^3*c^3*arcsin(c*x) + a*b^2*c^3)