Optimal. Leaf size=10 \[ -\frac {\text {Si}(\text {ArcCos}(a x))}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4720, 3380}
\begin {gather*} -\frac {\text {Si}(\text {ArcCos}(a x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4720
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} -\frac {\text {Si}(\text {ArcCos}(a x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 11, normalized size = 1.10
method | result | size |
derivativedivides | \(-\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{a}\) | \(11\) |
default | \(-\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{a}\) | \(11\) |
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 {1}{\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 = 10, normalized size = 1.00 \begin {gather*} -\frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{a} \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 = -0.10 \begin {gather*} \int \frac {1}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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