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=59 \[ 2 \text {Int}\left (\frac {a+b x}{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)},x\right )-\frac {1}{b \left (1-(a+b x)^2\right ) \sin ^{-1}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^2} \, dx \]

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*}

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Mathematica [A]  time = 11.90, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^2} \, dx \]

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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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \arcsin \left (b x + a\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}}\,{d x} \]

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)

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maple [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}} \arcsin \left (b x +a \right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + {\left (a^{2} - 1\right )} b\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right ) \int \frac {b x + a}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}\,{d x} + 1}{{\left (b^{3} x^{2} + 2 \, a b^{2} x + {\left (a^{2} - 1\right )} b\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asin}\left (a+b\,x\right )}^2\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]

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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