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

Optimal. Leaf size=41 \[ \text{Unintegrable}\left (\frac{\text{csch}^2\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{(1-a x) (a x+1)},x\right ) \]

[Out]

Unintegrable[Csch[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/((1 - a*x)*(1 + a*x)), x]

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Rubi [A]  time = 0.0799666, 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 = {} \[ \int \frac{\text{csch}^2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 36.221, size = 0, normalized size = 0. \[ \int \frac{\text{csch}^2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ( \sinh \left ({\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*x + 1)/(sqrt(-a*x + 1)*a*e^(2*sqrt(-a*x + 1)/sqrt(a*x + 1)) - sqrt(-a*x + 1)*a) - integrate(sqrt(a*x
+ 1)/((a^2*x^2 - 1)*sqrt(-a*x + 1)*e^(sqrt(-a*x + 1)/sqrt(a*x + 1)) + (a^2*x^2 - 1)*sqrt(-a*x + 1)), x) + inte
grate(sqrt(a*x + 1)/((a^2*x^2 - 1)*sqrt(-a*x + 1)*e^(sqrt(-a*x + 1)/sqrt(a*x + 1)) - (a^2*x^2 - 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 )} \sinh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )} \sinh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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