Optimal. Leaf size=59 \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.09, antiderivative size = 59, 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 = {4622, 4724, 3304, 3352} \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4622
Rule 4724
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}+(2 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {4 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 86, normalized size = 1.46 \[ -\frac {-2 \sqrt {1-a^2 x^2}-i \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )+i \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )}{a \sqrt {\cos ^{-1}(a x)}} \]
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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\arccos \left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 66, normalized size = 1.12 \[ -\frac {\sqrt {2}\, \left (2 \arccos \left (a x \right ) \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{a \sqrt {\pi }\, \arccos \left (a x \right )} \]
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 = -0.02 \[ \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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