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=43 \[ \text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 5.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]

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]

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fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}, x\right ) \]

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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, choosing root of [1,0,-4,0,%%%{4,[2,2]%%%}] at parameters values [86,-97]Warning, choosing root of [1,0,-
4,0,%%%{4,[2,2]%%%}] at parameters values [-82,7]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur
& l) Error: Bad Argument Value

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maple [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \,\mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{2}}\, dx \]

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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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