3.4.77 \(\int \frac {1}{(c-a^2 c x^2) \text {ArcSin}(a x)^2} \, dx\) [377]

Optimal. Leaf size=59 \[ -\frac {1}{a c \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}+\frac {a \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)},x\right )}{c} \]

[Out]

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

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Rubi [A]
time = 0.07, 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 = {} \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right ) \text {ArcSin}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right ) \text {ArcSin}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right ) \arcsin \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asin}^{2}{\left (a x \right )} - \operatorname {asin}^{2}{\left (a x \right )}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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