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

Optimal. Leaf size=39 \[ \text{Unintegrable}\left (\frac{\text{sech}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{(1-a x) (a x+1)},x\right ) \]

[Out]

Unintegrable[Sech[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/((1 - a*x)*(1 + a*x)), x]

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Rubi [A]  time = 0.0375866, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{sech}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\text{sech}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\text{sech}(x)}{x} \, dx,x,\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 5.70574, size = 0, normalized size = 0. \[ \int \frac{\text{sech}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ( \cosh \left ({\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)