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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 54.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x*(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 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{4} x^{5} - 2 \, a^{2} c^{2} x^{3} + a^{2} x + {\left (b^{2} c^{4} x^{5} - 2 \, b^{2} c^{2} x^{3} + b^{2} x\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} x^{5} - 2 \, a b c^{2} x^{3} + a b x\right )} \arcsin \left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-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^5 - 2*a^2*c^2*x^3 + a^2*x + (b^2*c^4*x^5 - 2*b^2*c^2*x^3 + b^2*x)*arcsi
n(c*x)^2 + 2*(a*b*c^4*x^5 - 2*a*b*c^2*x^3 + a*b*x)*arcsin(c*x)), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 3.83, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(-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 \[ \frac {\frac {{\left (a b c^{3} x^{3} - a b c x + {\left (b^{2} c^{3} x^{3} - b^{2} c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} {\left (3 \, c^{2} \int \frac {x^{2}}{b c^{4} x^{6} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a c^{4} x^{6} - 2 \, b c^{2} x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - 2 \, a c^{2} x^{4} + b x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a x^{2}}\,{d x} - \int \frac {1}{b c^{4} x^{6} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a c^{4} x^{6} - 2 \, b c^{2} x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - 2 \, a c^{2} x^{4} + b x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a x^{2}}\,{d x}\right )}}{b c} + 1}{a b c^{3} x^{3} - a b c x + {\left (b^{2} c^{3} x^{3} - b^{2} c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*b*c^3*x^3 - a*b*c*x + (b^2*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*integrate((3*c^2
*x^2 - 1)/(a*b*c^5*x^6 - 2*a*b*c^3*x^4 + a*b*c*x^2 + (b^2*c^5*x^6 - 2*b^2*c^3*x^4 + b^2*c*x^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1))), x) + 1)/(a*b*c^3*x^3 - a*b*c*x + (b^2*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1)))

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x*(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 \[ \int \frac {1}{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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