\(\int \frac {\text {sech}(\frac {\sqrt {1-a x}}{\sqrt {1+a x}})}{1-a^2 x^2} \, dx\) [216]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\text {Int}\left (\frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{(1-a x) (1+a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.83 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.98 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {1}{a^{2} x^{2} \cosh {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \cosh {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\text {sech}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {1}{\mathrm {cosh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )\,\left (a^2\,x^2-1\right )} \,d x \]

[In]

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

[Out]

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