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

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

[Out]

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

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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 = {} \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {x^m \sqrt {1-c^2 x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{m}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)*x^m/(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2), 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(x^m*(-c^2*x^2+1)^(1/2)/(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 = 0.93, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {-c^{2} x^{2}+1}}{\left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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