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

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

[Out]

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

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Rubi [A]  time = 0.0451338, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Mathematica [A]  time = 0.774385, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b{\it Arcsinh} \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*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

[Out]

int(1/(-c^2*x^2+1)/(a+b*arcsinh((-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 \,{\left (\sqrt{2} + \sqrt{-c x + 1}\right )}}{2 \, \sqrt{2} a b c^{2} x - 2 \, \sqrt{2} a b c - 4 \, \sqrt{-c x + 1} a b c -{\left (\sqrt{2} b^{2} c^{2} x - \sqrt{2} b^{2} c - 2 \, \sqrt{-c x + 1} b^{2} c\right )} \log \left (c x + 1\right ) + 2 \,{\left (\sqrt{2} b^{2} c^{2} x - \sqrt{2} b^{2} c - 2 \, \sqrt{-c x + 1} b^{2} c\right )} \log \left (\sqrt{2} + \sqrt{-c x + 1}\right )} - \int \frac{4 \, c x +{\left (\sqrt{2} c x - 3 \, \sqrt{2}\right )} \sqrt{-c x + 1} - 4}{2 \, a b c^{3} x^{3} - 6 \, a b c^{2} x^{2} + 6 \, a b c x - 4 \,{\left (a b c x - a b\right )}{\left (c x - 1\right )} - 2 \, a b -{\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x - 2 \,{\left (b^{2} c x - b^{2}\right )}{\left (c x - 1\right )} - b^{2} - 2 \,{\left (\sqrt{2} b^{2} c^{2} x^{2} - 2 \, \sqrt{2} b^{2} c x + \sqrt{2} b^{2}\right )} \sqrt{-c x + 1}\right )} \log \left (c x + 1\right ) + 2 \,{\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x - 2 \,{\left (b^{2} c x - b^{2}\right )}{\left (c x - 1\right )} - b^{2} - 2 \,{\left (\sqrt{2} b^{2} c^{2} x^{2} - 2 \, \sqrt{2} b^{2} c x + \sqrt{2} b^{2}\right )} \sqrt{-c x + 1}\right )} \log \left (\sqrt{2} + \sqrt{-c x + 1}\right ) - 4 \,{\left (\sqrt{2} a b c^{2} x^{2} - 2 \, \sqrt{2} a b c x + \sqrt{2} a b\right )} \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*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="maxima")

[Out]

-4*(sqrt(2) + sqrt(-c*x + 1))/(2*sqrt(2)*a*b*c^2*x - 2*sqrt(2)*a*b*c - 4*sqrt(-c*x + 1)*a*b*c - (sqrt(2)*b^2*c
^2*x - sqrt(2)*b^2*c - 2*sqrt(-c*x + 1)*b^2*c)*log(c*x + 1) + 2*(sqrt(2)*b^2*c^2*x - sqrt(2)*b^2*c - 2*sqrt(-c
*x + 1)*b^2*c)*log(sqrt(2) + sqrt(-c*x + 1))) - integrate((4*c*x + (sqrt(2)*c*x - 3*sqrt(2))*sqrt(-c*x + 1) -
4)/(2*a*b*c^3*x^3 - 6*a*b*c^2*x^2 + 6*a*b*c*x - 4*(a*b*c*x - a*b)*(c*x - 1) - 2*a*b - (b^2*c^3*x^3 - 3*b^2*c^2
*x^2 + 3*b^2*c*x - 2*(b^2*c*x - b^2)*(c*x - 1) - b^2 - 2*(sqrt(2)*b^2*c^2*x^2 - 2*sqrt(2)*b^2*c*x + sqrt(2)*b^
2)*sqrt(-c*x + 1))*log(c*x + 1) + 2*(b^2*c^3*x^3 - 3*b^2*c^2*x^2 + 3*b^2*c*x - 2*(b^2*c*x - b^2)*(c*x - 1) - b
^2 - 2*(sqrt(2)*b^2*c^2*x^2 - 2*sqrt(2)*b^2*c*x + sqrt(2)*b^2)*sqrt(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1)) -
4*(sqrt(2)*a*b*c^2*x^2 - 2*sqrt(2)*a*b*c*x + sqrt(2)*a*b)*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{arsinh}\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{arsinh}\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*arcsinh((-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)*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2
- a*b)*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1))), 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/(-c**2*x**2+1)/(a+b*asinh((-c*x+1)**(1/2)/(c*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 (c^{2} x^{2} - 1\right )}{\left (b \operatorname{arsinh}\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*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="giac")

[Out]

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