\(\int \frac {1}{(1-c^2 x^2)^{5/2} (a+b \arcsin (c x))^2} \, dx\) [427]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=-\frac {1}{b c \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}+\frac {4 c \text {Int}\left (\frac {x}{\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))},x\right )}{b} \]

[Out]

-1/b/c/(-c^2*x^2+1)^2/(a+b*arcsin(c*x))+4*c*Unintegrable(x/(-c^2*x^2+1)^3/(a+b*arcsin(c*x)),x)/b

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

Int[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

-(1/(b*c*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))) + (4*c*Defer[Int][x/((1 - c^2*x^2)^3*(a + b*ArcSin[c*x])), x])/
b

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b c \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}+\frac {(4 c) \int \frac {x}{\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))} \, dx}{b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]

[In]

int(1/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x)

[Out]

int(1/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.64 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)/(a^2*c^6*x^6 - 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 - 3*b^2*c^4*x^4 + 3*b
^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - a^2 + 2*(a*b*c^6*x^6 - 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 - a*b)*arcsin(c*x)), x)

Sympy [N/A]

Not integrable

Time = 3.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(1/((-(c*x - 1)*(c*x + 1))**(5/2)*(a + b*asin(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 255, normalized size of antiderivative = 10.20 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-(4*(a*b*c^6*x^4 - 2*a*b*c^4*x^2 + a*b*c^2 + (b^2*c^6*x^4 - 2*b^2*c^4*x^2 + b^2*c^2)*arctan2(c*x, sqrt(c*x + 1
)*sqrt(-c*x + 1)))*integrate(x/(a*b*c^6*x^6 - 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 - a*b + (b^2*c^6*x^6 - 3*b^2*c^4*x
^4 + 3*b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) + 1)/(a*b*c^5*x^4 - 2*a*b*c^3*x^2 +
a*b*c + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

Giac [N/A]

Not integrable

Time = 2.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((-c^2*x^2 + 1)^(5/2)*(b*arcsin(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int(1/((a + b*asin(c*x))^2*(1 - c^2*x^2)^(5/2)),x)

[Out]

int(1/((a + b*asin(c*x))^2*(1 - c^2*x^2)^(5/2)), x)