\(\int \frac {\sec ^2(\frac {\sqrt {1-a x}}{\sqrt {1+a x}})}{1-a^2 x^2} \, dx\) [57]

Optimal result

Integrand size = 36, antiderivative size = 36 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\text {Int}\left (\frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{(1-a x) (1+a x)},x\right ) \]

[Out]

Unintegrable(sec((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2/(-a*x+1)/(a*x+1),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 36, 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 = {} \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

[In]

Int[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]

[Out]

-(Defer[Subst][Defer[Int][Sec[x]^2/x, x], x, Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sec ^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 13.94 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

[In]

Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]

[Out]

Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89

\[\int \frac {1}{\left (-a^{2} x^{2}+1\right ) \cos \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{2}}d x\]

[In]

int(1/(-a^2*x^2+1)/cos((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2,x)

[Out]

int(1/(-a^2*x^2+1)/cos((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]

[In]

integrate(1/(-a^2*x^2+1)/cos((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2,x, algorithm="fricas")

[Out]

integral(-1/((a^2*x^2 - 1)*cos(sqrt(-a*x + 1)/sqrt(a*x + 1))^2), x)

Sympy [N/A]

Not integrable

Time = 21.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {1}{a^{2} x^{2} \cos ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \cos ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]

[In]

integrate(1/(-a**2*x**2+1)/cos((-a*x+1)**(1/2)/(a*x+1)**(1/2))**2,x)

[Out]

-Integral(1/(a**2*x**2*cos(sqrt(-a*x + 1)/sqrt(a*x + 1))**2 - cos(sqrt(-a*x + 1)/sqrt(a*x + 1))**2), x)

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 420, normalized size of antiderivative = 11.67 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]

[In]

integrate(1/(-a^2*x^2+1)/cos((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2,x, algorithm="maxima")

[Out]

2*((a^2*x + (a^2*x - a)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1))^2 + (a^2*x - a)*sin(2*sqrt(-a*x + 1)/sqrt(a*x + 1)
)^2 + 2*(a^2*x - a)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1)) - a)*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*sin(2*sqrt
(-a*x + 1)/sqrt(a*x + 1))/(a^3*x^3 - a^2*x^2 + (a^3*x^3 - a^2*x^2 - a*x + 1)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1
))^2 + (a^3*x^3 - a^2*x^2 - a*x + 1)*sin(2*sqrt(-a*x + 1)/sqrt(a*x + 1))^2 - a*x + 2*(a^3*x^3 - a^2*x^2 - a*x
+ 1)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1)) + 1), x) + sqrt(a*x + 1)*sqrt(-a*x + 1)*sin(2*sqrt(-a*x + 1)/sqrt(a*x
+ 1)))/(a^2*x + (a^2*x - a)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1))^2 + (a^2*x - a)*sin(2*sqrt(-a*x + 1)/sqrt(a*x
+ 1))^2 + 2*(a^2*x - a)*cos(2*sqrt(-a*x + 1)/sqrt(a*x + 1)) - a)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]

[In]

integrate(1/(-a^2*x^2+1)/cos((-a*x+1)^(1/2)/(a*x+1)^(1/2))^2,x, algorithm="giac")

[Out]

integrate(-1/((a^2*x^2 - 1)*cos(sqrt(-a*x + 1)/sqrt(a*x + 1))^2), x)

Mupad [N/A]

Not integrable

Time = 25.94 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {1}{{\cos \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]

[In]

int(-1/(cos((1 - a*x)^(1/2)/(a*x + 1)^(1/2))^2*(a^2*x^2 - 1)),x)

[Out]

-int(1/(cos((1 - a*x)^(1/2)/(a*x + 1)^(1/2))^2*(a^2*x^2 - 1)), x)