Optimal. Leaf size=91 \[ \frac{i b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{6 \left (-i d x^3\right )^{2/3}}-\frac{i b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{6 \left (i d x^3\right )^{2/3}}+\frac{a x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0642006, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {14, 3389, 2218} \[ \frac{i b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{6 \left (-i d x^3\right )^{2/3}}-\frac{i b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{6 \left (i d x^3\right )^{2/3}}+\frac{a x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int x \left (a+b \sin \left (c+d x^3\right )\right ) \, dx &=\int \left (a x+b x \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \sin \left (c+d x^3\right ) \, dx\\ &=\frac{a x^2}{2}+\frac{1}{2} (i b) \int e^{-i c-i d x^3} x \, dx-\frac{1}{2} (i b) \int e^{i c+i d x^3} x \, dx\\ &=\frac{a x^2}{2}+\frac{i b e^{i c} x^2 \Gamma \left (\frac{2}{3},-i d x^3\right )}{6 \left (-i d x^3\right )^{2/3}}-\frac{i b e^{-i c} x^2 \Gamma \left (\frac{2}{3},i d x^3\right )}{6 \left (i d x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.12647, size = 108, normalized size = 1.19 \[ \frac{x^2 \left (b \left (-i d x^3\right )^{2/3} (-\sin (c)-i \cos (c)) \text{Gamma}\left (\frac{2}{3},i d x^3\right )+i b \left (i d x^3\right )^{2/3} (\cos (c)+i \sin (c)) \text{Gamma}\left (\frac{2}{3},-i d x^3\right )+3 a \left (d^2 x^6\right )^{2/3}\right )}{6 \left (d^2 x^6\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.15415, size = 375, normalized size = 4.12 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{\left (x^{3}{\left | d \right |}\right )^{\frac{1}{3}}{\left ({\left ({\left (-i \, \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) +{\left (-i \, \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) -{\left (\Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left ({\left (\Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) -{\left (i \, \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) - i \, \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right ) -{\left (-i \, \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{1}{3} \, \pi + \frac{2}{3} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} b}{12 \, x{\left | d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73832, size = 149, normalized size = 1.64 \begin{align*} \frac{3 \, a d x^{2} - b \left (i \, d\right )^{\frac{1}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) - b \left (-i \, d\right )^{\frac{1}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \sin{\left (c + d x^{3} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x^{3} + c\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]