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