Optimal. Leaf size=51 \[ \frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{2 a}+\frac {x}{2 \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4622, 4720, 4624, 3299} \[ \frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{2 a}+\frac {x}{2 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4622
Rule 4624
Rule 4720
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)^3} \, dx &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}-\frac {1}{2} \int \frac {1}{\cos ^{-1}(a x)} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.92 \[ \frac {\sqrt {1-a^2 x^2}+\cos ^{-1}(a x)^2 \text {Si}\left (\cos ^{-1}(a x)\right )+a x \cos ^{-1}(a x)}{2 a \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arccos \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 43, normalized size = 0.84 \[ \frac {x}{2 \, \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{2 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a \arccos \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.84 \[ \frac {\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arccos \left (a x \right )^{2}}+\frac {a x}{2 \arccos \left (a x \right )}+\frac {\Si \left (\arccos \left (a x \right )\right )}{2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \int \frac {1}{\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} - a x \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right ) - \sqrt {a x + 1} \sqrt {-a x + 1}}{2 \, a \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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