Optimal. Leaf size=38 \[ \frac {x \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {\text {CosIntegral}(2 \text {ArcCos}(a x))}{a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3383}
\begin {gather*} \frac {x \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {\text {CosIntegral}(2 \text {ArcCos}(a x))}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 4728
Rubi steps
\begin {align*} \int \frac {x}{\cos ^{-1}(a x)^2} \, dx &=\frac {x \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {\text {Ci}\left (2 \cos ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 37, normalized size = 0.97 \begin {gather*} \frac {\frac {a x \sqrt {1-a^2 x^2}}{\text {ArcCos}(a x)}-\text {CosIntegral}(2 \text {ArcCos}(a x))}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 30, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}-\cosineIntegral \left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(30\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}-\cosineIntegral \left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(30\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 36, normalized size = 0.95 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x}{a \arccos \left (a x\right )} - \frac {\operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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