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

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

[Out]

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

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Rubi [A]  time = 0.0426659, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcCoth[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 \coth ^{-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 \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.751716, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.976, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b{\rm arccoth} \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*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

[Out]

int(1/(-c^2*x^2+1)/(a+b*arccoth((-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{4 \, c x}{\sqrt{c x + 1} \sqrt{-c x + 1} b^{2} c \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) - \sqrt{c x + 1} \sqrt{-c x + 1} b^{2} c \log \left (-\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) + 2 \, \sqrt{c x + 1} \sqrt{-c x + 1} a b c} - \int -\frac{4}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \log \left (-\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)*b^2*c*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - sqrt(c*x + 1)*sqrt(-c*x + 1)*b
^2*c*log(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + 2*sqrt(c*x + 1)*sqrt(-c*x + 1)*a*b*c) - integrate(-4/((b^2*c^2*x^2
- b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - (b^2*c^2*x^2 - b^2)*sqrt(c*x + 1)*s
qrt(-c*x + 1)*log(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + 2*(a*b*c^2*x^2 - a*b)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)

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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 )} \operatorname{arcoth}\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 )} \operatorname{arcoth}\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*arccoth((-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)*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2
- a*b)*arccoth(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{acoth}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - 2 a b \operatorname{acoth}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} + b^{2} c^{2} x^{2} \operatorname{acoth}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - b^{2} \operatorname{acoth}^{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*acoth((-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*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1)) - 2*a*b*acoth(sqrt(-c
*x + 1)/sqrt(c*x + 1)) + b**2*c**2*x**2*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))**2 - b**2*acoth(sqrt(-c*x + 1)/sqr
t(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 \operatorname{arcoth}\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*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="giac")

[Out]

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