3.680 \(\int \frac {1}{(d+e x^2) (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 22.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e x^{2} + a^{2} d + {\left (b^{2} e x^{2} + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b e x^{2} + a b d\right )} \arcsin \left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a^2*e*x^2 + a^2*d + (b^2*e*x^2 + b^2*d)*arcsin(c*x)^2 + 2*(a*b*e*x^2 + a*b*d)*arcsin(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((e*x^2 + d)*(b*arcsin(c*x) + a)^2), x)

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maple [A]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)/(a+b*arcsin(c*x))^2,x)

[Out]

int(1/(e*x^2+d)/(a+b*arcsin(c*x))^2,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*asin(c*x))^2*(d + e*x^2)),x)

[Out]

int(1/((a + b*asin(c*x))^2*(d + e*x^2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)/(a+b*asin(c*x))**2,x)

[Out]

Integral(1/((a + b*asin(c*x))**2*(d + e*x**2)), x)

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