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

Optimal. Leaf size=64 \[ \frac {2 c \text {Int}\left (\frac {x}{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac {1}{b c \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{4} x^{4} - 2 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {2 \, {\left (a b c^{4} x^{2} - a b c^{2} + {\left (b^{2} c^{4} x^{2} - b^{2} c^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \int \frac {x}{{\left (c x + 1\right )}^{2} {\left (c x - 1\right )}^{2} {\left (b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a\right )}}\,{d x}}{b} + 1}{a b c^{3} x^{2} - a b c + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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