\(\int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx\) [683]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 15.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int \frac {1}{\sqrt {e \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(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)

Sympy [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \sqrt {d + e x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.58 (sec) , antiderivative size = 364, normalized size of antiderivative = 16.55 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

((a*b*c^3*d^2 + a*b*c*d*e + (a*b*c^3*d*e + a*b*c*e^2)*x^2 + (b^2*c^3*d^2 + b^2*c*d*e + (b^2*c^3*d*e + b^2*c*e^
2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*integrate(sqrt(e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x/(
a*b*c^3*e^2*x^6 - a*b*c*d^2 + (2*a*b*c^3*d*e - a*b*c*e^2)*x^4 + (a*b*c^3*d^2 - 2*a*b*c*d*e)*x^2 + (b^2*c^3*e^2
*x^6 - b^2*c*d^2 + (2*b^2*c^3*d*e - b^2*c*e^2)*x^4 + (b^2*c^3*d^2 - 2*b^2*c*d*e)*x^2)*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1))), x) - sqrt(e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x + 1))/(a*b*c*e*x^2 + a*b*c*d + (b^2*c*e*x^2
+ b^2*c*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e\,x^2+d}} \,d x \]

[In]

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

[Out]

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