Optimal. Leaf size=112 \[ -\frac{b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac{b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac{a x^5}{5}-\frac{b x^2 \cos \left (c+d x^3\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.085472, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {14, 3385, 3390, 2218} \[ -\frac{b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac{b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac{a x^5}{5}-\frac{b x^2 \cos \left (c+d x^3\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 3385
Rule 3390
Rule 2218
Rubi steps
\begin{align*} \int x^4 \left (a+b \sin \left (c+d x^3\right )\right ) \, dx &=\int \left (a x^4+b x^4 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac{a x^5}{5}+b \int x^4 \sin \left (c+d x^3\right ) \, dx\\ &=\frac{a x^5}{5}-\frac{b x^2 \cos \left (c+d x^3\right )}{3 d}+\frac{(2 b) \int x \cos \left (c+d x^3\right ) \, dx}{3 d}\\ &=\frac{a x^5}{5}-\frac{b x^2 \cos \left (c+d x^3\right )}{3 d}+\frac{b \int e^{-i c-i d x^3} x \, dx}{3 d}+\frac{b \int e^{i c+i d x^3} x \, dx}{3 d}\\ &=\frac{a x^5}{5}-\frac{b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac{b e^{i c} x^2 \Gamma \left (\frac{2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac{b e^{-i c} x^2 \Gamma \left (\frac{2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.227819, size = 124, normalized size = 1.11 \[ \frac{d x^8 \left (-5 b \left (-i d x^3\right )^{2/3} (\cos (c)-i \sin (c)) \text{Gamma}\left (\frac{2}{3},i d x^3\right )-5 b \left (i d x^3\right )^{2/3} (\cos (c)+i \sin (c)) \text{Gamma}\left (\frac{2}{3},-i d x^3\right )+3 \left (d^2 x^6\right )^{2/3} \left (3 a d x^3-5 b \cos \left (c+d x^3\right )\right )\right )}{45 \left (d^2 x^6\right )^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \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 = 1.17716, size = 402, normalized size = 3.59 \begin{align*} \frac{1}{5} \, a x^{5} - \frac{{\left (6 \, x^{3}{\left | d \right |} \cos \left (d x^{3} + c\right ) + \left (x^{3}{\left | d \right |}\right )^{\frac{1}{3}}{\left ({\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 )} \cos \left (c\right ) -{\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 )} \sin \left (c\right )\right )}\right )} b}{18 \, d x{\left | d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6596, size = 204, normalized size = 1.82 \begin{align*} \frac{9 \, a d^{2} x^{5} - 15 \, b d x^{2} \cos \left (d x^{3} + c\right ) + 5 i \, b \left (i \, d\right )^{\frac{1}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) - 5 i \, b \left (-i \, d\right )^{\frac{1}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right )}{45 \, d^{2}} \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^{4} \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^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]