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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0436054, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Mathematica [A]  time = 5.09053, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b{\rm arccosh} \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*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*c*x*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + (c*x + 1)*sqrt(-c*x + 1)
- (-c*x + 1)^(3/2))/(2*(c*x + 1)*sqrt(-c*x + 1)*a*b*c - 2*(-c*x + 1)^(3/2)*a*b*c - ((c*x - 1)*b^2*c*log(c*x +
1) - 2*(c*x - 1)*a*b*c)*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) - ((c*x +
1)*sqrt(-c*x + 1)*b^2*c - (-c*x + 1)^(3/2)*b^2*c)*log(c*x + 1) + 2*((c*x - 1)*b^2*c*sqrt(sqrt(c*x + 1) + sqrt(
-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + (c*x + 1)*sqrt(-c*x + 1)*b^2*c - (-c*x + 1)^(3/2)*b^2*c)*lo
g(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))) - integrate(-2
*(2*(c*x + 1)*sqrt(-c*x + 1)*(sqrt(c*x + 1) + sqrt(-c*x + 1))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) + ((c*x + 1)^2
+ 2*(c*x + 1)*(c*x - 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)))/(2*(a*b*c
^2*x^2 - a*b)*(c*x + 1)^2*sqrt(-c*x + 1) - 4*(a*b*c^2*x^2 - a*b)*(c*x + 1)*(-c*x + 1)^(3/2) + 2*(a*b*c^2*x^2 -
a*b)*(-c*x + 1)^(5/2) + ((b^2*c^2*x^2 - b^2)*(-c*x + 1)^(3/2)*log(c*x + 1) - 2*(a*b*c^2*x^2 - a*b)*(-c*x + 1)
^(3/2))*(sqrt(c*x + 1) + sqrt(-c*x + 1))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) + 2*(2*(a*b*c^2*x^2 - a*b)*(c*x + 1)
*(c*x - 1) + 2*(a*b*c^2*x^2 - a*b)*(c*x - 1)^2 - ((b^2*c^2*x^2 - b^2)*(c*x + 1)*(c*x - 1) + (b^2*c^2*x^2 - b^2
)*(c*x - 1)^2)*log(c*x + 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) - ((b^
2*c^2*x^2 - b^2)*(c*x + 1)^2*sqrt(-c*x + 1) - 2*(b^2*c^2*x^2 - b^2)*(c*x + 1)*(-c*x + 1)^(3/2) + (b^2*c^2*x^2
- b^2)*(-c*x + 1)^(5/2))*log(c*x + 1) - 2*((b^2*c^2*x^2 - b^2)*(-c*x + 1)^(3/2)*(sqrt(c*x + 1) + sqrt(-c*x + 1
))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) - (b^2*c^2*x^2 - b^2)*(c*x + 1)^2*sqrt(-c*x + 1) + 2*(b^2*c^2*x^2 - b^2)*(
c*x + 1)*(-c*x + 1)^(3/2) - (b^2*c^2*x^2 - b^2)*(-c*x + 1)^(5/2) - 2*((b^2*c^2*x^2 - b^2)*(c*x + 1)*(c*x - 1)
+ (b^2*c^2*x^2 - b^2)*(c*x - 1)^2)*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)))
*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))), x)

________________________________________________________________________________________

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{arcosh}\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{arcosh}\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*arccosh((-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)*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2
- a*b)*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1))), x)

________________________________________________________________________________________

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*acosh((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="giac")

[Out]

Timed out