3.36 \(\int \frac{1}{(1-c^2 x^2) (a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0411422, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx &=\int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.778394, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b\arctan \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-c^2*x^2+1)/(a+b*arctan((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

[Out]

int(1/(-c^2*x^2+1)/(a+b*arctan((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*(b^2*c^2*arctan2(sqrt(-c*x + 1), sqrt(c*x + 1)) + a*b*c^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*integrate(1/2*x/(
(a*b*c^2*x^2 - a*b + (b^2*c^2*x^2 - b^2)*arctan2(sqrt(-c*x + 1), sqrt(c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))
, x) + 1)/((b^2*c*arctan2(sqrt(-c*x + 1), sqrt(c*x + 1)) + a*b*c)*sqrt(c*x + 1)*sqrt(-c*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arctan(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2 -
a*b)*arctan(sqrt(-c*x + 1)/sqrt(c*x + 1))), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a^{2} c^{2} x^{2} - a^{2} + 2 a b c^{2} x^{2} \operatorname{atan}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - 2 a b \operatorname{atan}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} + b^{2} c^{2} x^{2} \operatorname{atan}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - b^{2} \operatorname{atan}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c**2*x**2+1)/(a+b*atan((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2,x)

[Out]

-Integral(1/(a**2*c**2*x**2 - a**2 + 2*a*b*c**2*x**2*atan(sqrt(-c*x + 1)/sqrt(c*x + 1)) - 2*a*b*atan(sqrt(-c*x
+ 1)/sqrt(c*x + 1)) + b**2*c**2*x**2*atan(sqrt(-c*x + 1)/sqrt(c*x + 1))**2 - b**2*atan(sqrt(-c*x + 1)/sqrt(c*
x + 1))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-1/((c^2*x^2 - 1)*(b*arctan(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2), x)