3.2.6 \(\int \frac {1}{(1-c^2 x^2) (a+b \text {ArcCos}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2} \, dx\) [106]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {ArcCos}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Mathematica [A]
time = 1.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {ArcCos}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(-c^2*x^2+1)/(a+b*arccos((-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((sqrt(2)*b^2*c*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + sqrt(2)*a*b*c - (sqrt(2)*b^2*c^2*arctan2(sq
rt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + sqrt(2)*a*b*c^2)*x)*sqrt(c)*integrate(1/2*sqrt(-c*x + 1)*sqrt(x)/((b^
2*c^3*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + a*b*c^3)*x^3 - 2*(b^2*c^2*arctan2(sqrt(2)*sqrt(c)*sqr
t(x), sqrt(-c*x + 1)) + a*b*c^2)*x^2 + (b^2*c*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + a*b*c)*x), x)
- sqrt(2)*sqrt(-c*x + 1)*sqrt(c)*sqrt(x))/(b^2*c*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + a*b*c - (
b^2*c^2*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1)) + a*b*c^2)*x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c**2*x**2+1)/(a+b*acos((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2,x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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