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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{\sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0253785, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 8.37607, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}}{b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + a^{2} d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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