Optimal. Leaf size=31 \[ -\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4720, 3386,
3432} \begin {gather*} -\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rule 4720
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a}\\ &=-\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.02, size = 68, normalized size = 2.19 \begin {gather*} -\frac {-\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )-\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )}{2 a \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 26, normalized size = 0.84
method | result | size |
default | \(-\frac {\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{a}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 47, normalized size = 1.52 \begin {gather*} -\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4 \, a} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4 \, 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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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