Optimal. Leaf size=115 \[ \frac {i x^{3 a+1} \left (-i x^{2 a}\right )^{-\frac {3 a+1}{2 a}} \operatorname {Gamma}\left (\frac {1}{2} \left (\frac {1}{a}+3\right ),-i x^{2 a}\right )}{4 a}-\frac {i x^{3 a+1} \left (i x^{2 a}\right )^{-\frac {3 a+1}{2 a}} \operatorname {Gamma}\left (\frac {1}{2} \left (\frac {1}{a}+3\right ),i x^{2 a}\right )}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 2218
Rule 3423
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 = 0.30, size = 142, normalized size = 1.23 \[ -\frac {x^{a+1} \left (x^{4 a}\right )^{-\frac {a+1}{2 a}} \left ((a+1) \left (-i x^{2 a}\right )^{\frac {a+1}{2 a}} \operatorname {Gamma}\left (\frac {a+1}{2 a},i x^{2 a}\right )+(a+1) \left (i x^{2 a}\right )^{\frac {a+1}{2 a}} \operatorname {Gamma}\left (\frac {a+1}{2 a},-i x^{2 a}\right )+4 a \left (x^{4 a}\right )^{\frac {a+1}{2 a}} \cos \left (x^{2 a}\right )\right )}{8 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3 \, a} \sin \left (x^{2 \, a}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3 \, a} \sin \left (x^{2 \, a}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.16, size = 41, normalized size = 0.36 \[ \frac {x^{5 a +1} \hypergeom \left (\left [\frac {1}{4 a}+\frac {5}{4}\right ], \left [\frac {3}{2}, \frac {1}{4 a}+\frac {9}{4}\right ], -\frac {x^{4 a}}{4}\right )}{5 a +1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {x x^{a} \cos \left (x^{2 \, a}\right ) - \frac {{\left (a + 1\right )} x x^{a} \Gamma \left (\frac {1}{4 \, a} + \frac {1}{4}\right ) \,_1F_2\left (\begin {matrix} \frac {1}{4 \, a} + \frac {1}{4} \\ \frac {1}{2},\frac {1}{4 \, a} + \frac {5}{4} \end {matrix} ; -\frac {1}{4} \, x^{4 \, a} \right )}{4 \, a \Gamma \left (\frac {1}{4 \, a} + \frac {5}{4}\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{3\,a}\,\sin \left (x^{2\,a}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.89, size = 54, normalized size = 0.47 \[ \frac {x x^{5 a} \Gamma \left (\frac {5}{4} + \frac {1}{4 a}\right ) {{}_{1}F_{2}\left (\begin {matrix} \frac {5}{4} + \frac {1}{4 a} \\ \frac {3}{2}, \frac {9}{4} + \frac {1}{4 a} \end {matrix}\middle | {- \frac {x^{4 a}}{4}} \right )}}{4 a \Gamma \left (\frac {9}{4} + \frac {1}{4 a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________