3.361 \(\int \frac{1}{x^2 (1-c^2 x^2)^{3/2} (a+b \sin ^{-1}(c x))} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]

________________________________________________________________________________________

Rubi [A]  time = 0.136276, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]

[Out]

Defer[Int][1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 1.8818, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]

[Out]

Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]

________________________________________________________________________________________

Maple [A]  time = 0.385, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) } \left ( -{c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x)),x)

[Out]

int(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x)),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((-c^2*x^2 + 1)^(3/2)*(b*arcsin(c*x) + a)*x^2), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{a c^{4} x^{6} - 2 \, a c^{2} x^{4} + a x^{2} +{\left (b c^{4} x^{6} - 2 \, b c^{2} x^{4} + b x^{2}\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a*c^4*x^6 - 2*a*c^2*x^4 + a*x^2 + (b*c^4*x^6 - 2*b*c^2*x^4 + b*x^2)*arcsin(c*x)),
x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(-c**2*x**2+1)**(3/2)/(a+b*asin(c*x)),x)

[Out]

Integral(1/(x**2*(-(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate(1/((-c^2*x^2 + 1)^(3/2)*(b*arcsin(c*x) + a)*x^2), x)