Optimal. Leaf size=26 \[ -\sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(x)}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4724, 3304, 3352} \[ -\sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4724
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-x^2} \sqrt {\cos ^{-1}(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(x)}\right )\right )\\ &=-\sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(x)}\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 56, normalized size = 2.15 \[ \frac {i \left (\sqrt {-i \cos ^{-1}(x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(x)\right )-\sqrt {i \cos ^{-1}(x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(x)\right )\right )}{2 \sqrt {\cos ^{-1}(x)}} \]
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 = 0.28, size = 58, normalized size = 2.23 \[ \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \relax (x)}}{i - 1}\right )}{2 \, {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \relax (x)}}{i - 1}\right )}{2 \, {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 21, normalized size = 0.81 \[ -\FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \relax (x )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi } \]
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 = -0.04 \[ \int \frac {x}{\sqrt {\mathrm {acos}\relax (x)}\,\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {\operatorname {acos}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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