3.166 \(\int \frac {1}{x (a+b \cos ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{x \left (a+b \cos ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {1}{x \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*ArcCos[c*x])^2),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b \cos ^{-1}(c x)\right )^2} \, dx &=\int \frac {1}{x \left (a+b \cos ^{-1}(c x)\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 7.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x \arccos \left (c x\right )^{2} + 2 \, a b x \arccos \left (c x\right ) + a^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \arccos \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c
^3*x^4 - a*b*c*x^2 + (b^2*c^3*x^4 - b^2*c*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x) - sqrt(c*x + 1)
*sqrt(-c*x + 1))/(b^2*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*acos(c*x))^2),x)

[Out]

int(1/(x*(a + b*acos(c*x))^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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