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

Optimal. Leaf size=58 \[ 2 \text{Unintegrable}\left (\frac{a+b x}{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)},x\right )-\frac{1}{b \left (1-(a+b x)^2\right ) \sin ^{-1}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.124873, 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 = {} \[ \int \frac{1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 10.9414, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}} \left ( -{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \,{\left (3 \, a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \arcsin \left (b x + a\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/((b^4*x^4 + 4*a*b^3*x^3 + 2*(3*a^2 - 1)*b^2*x^2 + a^4 + 4*(a^3 - a
)*b*x - 2*a^2 + 1)*arcsin(b*x + a)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac{3}{2}} \operatorname{asin}^{2}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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