Optimal. Leaf size=115 \[ \frac{i x^{3 a+1} \left (-i x^{2 a}\right )^{-\frac{3 a+1}{2 a}} \text{Gamma}\left (\frac{1}{2} \left (\frac{1}{a}+3\right ),-i x^{2 a}\right )}{4 a}-\frac{i x^{3 a+1} \left (i x^{2 a}\right )^{-\frac{3 a+1}{2 a}} \text{Gamma}\left (\frac{1}{2} \left (\frac{1}{a}+3\right ),i x^{2 a}\right )}{4 a} \]
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Rubi [A] time = 0.0560074, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3423, 2218} \[ \frac{i x^{3 a+1} \left (-i x^{2 a}\right )^{-\frac{3 a+1}{2 a}} \text{Gamma}\left (\frac{1}{2} \left (\frac{1}{a}+3\right ),-i x^{2 a}\right )}{4 a}-\frac{i x^{3 a+1} \left (i x^{2 a}\right )^{-\frac{3 a+1}{2 a}} \text{Gamma}\left (\frac{1}{2} \left (\frac{1}{a}+3\right ),i x^{2 a}\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 3423
Rule 2218
Rubi steps
\begin{align*} \int x^{3 a} \sin \left (x^{2 a}\right ) \, dx &=\frac{1}{2} i \int e^{-i x^{2 a}} x^{3 a} \, dx-\frac{1}{2} i \int e^{i x^{2 a}} x^{3 a} \, dx\\ &=\frac{i x^{1+3 a} \left (-i x^{2 a}\right )^{-\frac{1+3 a}{2 a}} \Gamma \left (\frac{1}{2} \left (3+\frac{1}{a}\right ),-i x^{2 a}\right )}{4 a}-\frac{i x^{1+3 a} \left (i x^{2 a}\right )^{-\frac{1+3 a}{2 a}} \Gamma \left (\frac{1}{2} \left (3+\frac{1}{a}\right ),i x^{2 a}\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.293061, size = 142, normalized size = 1.23 \[ -\frac{x^{a+1} \left (x^{4 a}\right )^{-\frac{a+1}{2 a}} \left ((a+1) \left (-i x^{2 a}\right )^{\frac{a+1}{2 a}} \text{Gamma}\left (\frac{a+1}{2 a},i x^{2 a}\right )+(a+1) \left (i x^{2 a}\right )^{\frac{a+1}{2 a}} \text{Gamma}\left (\frac{a+1}{2 a},-i x^{2 a}\right )+4 a \left (x^{4 a}\right )^{\frac{a+1}{2 a}} \cos \left (x^{2 a}\right )\right )}{8 a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.089, size = 41, normalized size = 0.4 \begin{align*}{\frac{{x}^{5\,a+1}}{5\,a+1}{\mbox{$_1$F$_2$}({\frac{5}{4}}+{\frac{1}{4\,a}};\,{\frac{3}{2}},{\frac{9}{4}}+{\frac{1}{4\,a}};\,-{\frac{{x}^{4\,a}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x x^{a} \cos \left (x^{2 \, a}\right ) -{\left (a + 1\right )} \int x^{a} \cos \left (x^{2 \, a}\right )\,{d x}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3 \, a} \sin \left (x^{2 \, a}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.48852, size = 54, normalized size = 0.47 \begin{align*} \frac{x x^{5 a} \Gamma \left (\frac{5}{4} + \frac{1}{4 a}\right ){{}_{1}F_{2}\left (\begin{matrix} \frac{5}{4} + \frac{1}{4 a} \\ \frac{3}{2}, \frac{9}{4} + \frac{1}{4 a} \end{matrix}\middle |{- \frac{x^{4 a}}{4}} \right )}}{4 a \Gamma \left (\frac{9}{4} + \frac{1}{4 a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3 \, a} \sin \left (x^{2 \, a}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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