\(\int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx\) [26]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.96 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 5.11 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 426, normalized size of antiderivative = 23.67 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((a*b*c*e^2*x^2 + 2*a*b*c*d*e*x + a*b*c*d^2 + (b^2*c*e^2*x^2 + 2*b^2*c*d*e*x + b^2*c*d^2)*arctan2(c*x, sqrt(c
*x + 1)*sqrt(-c*x + 1)))*integrate((c^2*e*x^2 - c^2*d*x - 2*e)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*e^3*x^5 +
3*a*b*c^3*d*e^2*x^4 - 3*a*b*c*d^2*e*x - a*b*c*d^3 + (3*a*b*c^3*d^2*e - a*b*c*e^3)*x^3 + (a*b*c^3*d^3 - 3*a*b*
c*d*e^2)*x^2 + (b^2*c^3*e^3*x^5 + 3*b^2*c^3*d*e^2*x^4 - 3*b^2*c*d^2*e*x - b^2*c*d^3 + (3*b^2*c^3*d^2*e - b^2*c
*e^3)*x^3 + (b^2*c^3*d^3 - 3*b^2*c*d*e^2)*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^2 + 2*a*b*c*d*e*x + a*b*c*d^2 + (b^2*c*e^2*x^2 + 2*b^2*c*d*e*x + b^2*c*d^2)*arct
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

Giac [N/A]

Not integrable

Time = 0.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x)^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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