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

Optimal. Leaf size=42 \[ \text {Int}\left (\frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{(1-a x) (a x+1)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\text {sech}^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 {sech}^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ \end {align*}

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Mathematica [A]  time = 26.14, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}, x\right ) \]

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))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}\,{d x} \]

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))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} x^{2}+1\right ) \cosh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{2}}\, dx \]

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))^2,x)

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \sqrt {a x + 1}}{\sqrt {-a x + 1} a e^{\left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} + \sqrt {-a x + 1} a} + 2 \, \int \frac {\sqrt {a x + 1}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1} e^{\left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} + {\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1}}\,{d x} \]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{{\mathrm {cosh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{2} x^{2} \cosh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \cosh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]

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))**2,x)

[Out]

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

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