Optimal. Leaf size=71 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4406
Rule 4636
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 126, normalized size = 1.77 \[ -\frac {-3 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )-3 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )-\sqrt {3} \left (\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \cos ^{-1}(a x)\right )\right )}{24 a^3 \sqrt {\cos ^{-1}(a x)}} \]
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 = 0.34, size = 135, normalized size = 1.90 \[ \frac {\sqrt {6} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {6} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3} {\left (i - 1\right )}} + \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3} {\left (i - 1\right )}} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3} {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3} {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 50, normalized size = 0.70 \[ -\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+3 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{12 a^{3}} \]
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 = -0.01 \[ \int \frac {x^2}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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