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

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

[Out]

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

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Rubi [A]  time = 0.0436831, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )} \, dx \]

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]])),x]

[Out]

Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), 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 )} \, 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 )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0869805, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )} \, dx \]

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]])),x]

[Out]

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

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Maple [A]  time = 1.135, 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 ) ^{-1}}\, 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))),x)

[Out]

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

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Maxima [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 )}}\,{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))),x, algorithm="maxima")

[Out]

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

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

[Out]

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

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

[Out]

-Integral(1/(a*c**2*x**2 - a + b*c**2*x**2*atan(sqrt(-c*x + 1)/sqrt(c*x + 1)) - b*atan(sqrt(-c*x + 1)/sqrt(c*x
+ 1))), 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 )}}\,{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))),x, algorithm="giac")

[Out]

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