Optimal. Leaf size=63 \[ \frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{a^2}+\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)} \]
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Rubi [A] time = 0.16, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {4634, 4720, 4636, 4406, 12, 3299, 4642} \[ \frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{a^2}+\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 4634
Rule 4636
Rule 4642
Rule 4720
Rubi steps
\begin {align*} \int \frac {x}{\cos ^{-1}(a x)^3} \, dx &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{2 a}+a \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}-2 \int \frac {x}{\cos ^{-1}(a x)} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 1.00 \[ \frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{a^2}+\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {2 a^2 x^2-1}{2 a^2 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arccos \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 57, normalized size = 0.90 \[ \frac {x^{2}}{\arccos \left (a x\right )} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a \arccos \left (a x\right )^{2}} - \frac {1}{2 \, a^{2} \arccos \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.68 \[ \frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}+\Si \left (2 \arccos \left (a x \right )\right )}{a^{2}} \]
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 \[ -\frac {4 \, a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \int \frac {x}{\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} - \sqrt {a x + 1} \sqrt {-a x + 1} a x - {\left (2 \, a^{2} x^{2} - 1\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{2 \, a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}} \]
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 \[ \int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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