\(\int \frac {1}{(1-a^2-2 a b x-b^2 x^2)^{3/2} \arcsin (a+b x)^2} \, dx\) [337]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \left (1-(a+b x)^2\right ) \arcsin (a+b x)}+2 \text {Int}\left (\frac {a+b x}{\left (1-(a+b x)^2\right )^2 \arcsin (a+b x)},x\right ) \]

[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)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2} \arcsin (x)^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {1}{b \left (1-(a+b x)^2\right ) \arcsin (a+b x)}+\frac {2 \text {Subst}\left (\int \frac {x}{\left (1-x^2\right )^2 \arcsin (x)} \, dx,x,a+b x\right )}{b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.40 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 2.75 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94

\[\int \frac {1}{\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}} \arcsin \left (b x +a \right )^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.64 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int { \frac {1}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 3.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int \frac {1}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 6.53 (sec) , antiderivative size = 195, normalized size of antiderivative = 5.91 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int { \frac {1}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}} \,d x } \]

[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)))

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int { \frac {1}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2} \, dx=\int \frac {1}{{\mathrm {asin}\left (a+b\,x\right )}^2\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]

[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)