Optimal. Leaf size=31 \[ -\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a} \]
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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 = {4624, 3305, 3351} \[ -\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4624
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \operatorname {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] time = 0.03, size = 68, normalized size = 2.19 \[ -\frac {-\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )-\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )}{2 a \sqrt {\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.16, size = 68, normalized size = 2.19 \[ \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{2 \, a {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{2 \, a {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.84 \[ -\frac {\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{a} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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