Optimal. Leaf size=14 \[ -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4732, 4491, 12,
3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 4491
Rule 4732
Rubi steps
\begin {align*} \int \frac {x}{\cos ^{-1}(a x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 13, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
default | \(-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
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}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 12, normalized size = 0.86 \begin {gather*} -\frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, 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 = -0.07 \begin {gather*} \int \frac {x}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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