Optimal. Leaf size=65 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]
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Rubi [A] time = 0.08, antiderivative size = 65, 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} \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4406
Rule 4636
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^4}-\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^4}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 130, normalized size = 2.00 \[ -\frac {-2 \sqrt {2} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \cos ^{-1}(a x)\right )-2 \sqrt {2} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \cos ^{-1}(a x)\right )-\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \cos ^{-1}(a x)\right )-\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \cos ^{-1}(a x)\right )}{32 a^4 \sqrt {\cos ^{-1}(a x)}} \]
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 = 3.60, size = 113, normalized size = 1.74 \[ \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\sqrt {2} {\left (i - 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{4} {\left (i - 1\right )}} + \frac {\sqrt {\pi } i \operatorname {erf}\left ({\left (i - 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{4} {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} {\left (i + 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{4} {\left (i - 1\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-{\left (i + 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{4} {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 43, normalized size = 0.66 \[ -\frac {\sqrt {\pi }\, \left (\sqrt {2}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 a^{4}} \]
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 \[ \int \frac {x^3}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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