\(\int \frac {1}{(d+e x^2)^2 (a+b \arcsin (c x))^2} \, dx\) [681]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 53.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.90 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*
(a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*arcsin(c*x)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 2.63 (sec) , antiderivative size = 439, normalized size of antiderivative = 21.95 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((a*b*c*e^2*x^4 + 2*a*b*c*d*e*x^2 + a*b*c*d^2 + (b^2*c*e^2*x^4 + 2*b^2*c*d*e*x^2 + b^2*c*d^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1)))*integrate((3*c^2*e*x^3 - (c^2*d + 4*e)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*e
^3*x^8 + (3*a*b*c^3*d*e^2 - a*b*c*e^3)*x^6 - a*b*c*d^3 + 3*(a*b*c^3*d^2*e - a*b*c*d*e^2)*x^4 + (a*b*c^3*d^3 -
3*a*b*c*d^2*e)*x^2 + (b^2*c^3*e^3*x^8 + (3*b^2*c^3*d*e^2 - b^2*c*e^3)*x^6 - b^2*c*d^3 + 3*(b^2*c^3*d^2*e - b^2
*c*d*e^2)*x^4 + (b^2*c^3*d^3 - 3*b^2*c*d^2*e)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) + sqrt(c*x
+ 1)*sqrt(-c*x + 1))/(a*b*c*e^2*x^4 + 2*a*b*c*d*e*x^2 + a*b*c*d^2 + (b^2*c*e^2*x^4 + 2*b^2*c*d*e*x^2 + b^2*c*d
^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

Giac [N/A]

Not integrable

Time = 7.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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