Optimal. Leaf size=44 \[ \frac {a x^6}{6}+\frac {b \sin \left (c+d x^3\right )}{3 d^2}-\frac {b x^3 \cos \left (c+d x^3\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {14, 3379, 3296, 2637} \[ \frac {a x^6}{6}+\frac {b \sin \left (c+d x^3\right )}{3 d^2}-\frac {b x^3 \cos \left (c+d x^3\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2637
Rule 3296
Rule 3379
Rubi steps
\begin {align*} \int x^5 \left (a+b \sin \left (c+d x^3\right )\right ) \, dx &=\int \left (a x^5+b x^5 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {a x^6}{6}+b \int x^5 \sin \left (c+d x^3\right ) \, dx\\ &=\frac {a x^6}{6}+\frac {1}{3} b \operatorname {Subst}\left (\int x \sin (c+d x) \, dx,x,x^3\right )\\ &=\frac {a x^6}{6}-\frac {b x^3 \cos \left (c+d x^3\right )}{3 d}+\frac {b \operatorname {Subst}\left (\int \cos (c+d x) \, dx,x,x^3\right )}{3 d}\\ &=\frac {a x^6}{6}-\frac {b x^3 \cos \left (c+d x^3\right )}{3 d}+\frac {b \sin \left (c+d x^3\right )}{3 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 44, normalized size = 1.00 \[ \frac {a x^6}{6}+\frac {b \sin \left (c+d x^3\right )}{3 d^2}-\frac {b x^3 \cos \left (c+d x^3\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 40, normalized size = 0.91 \[ \frac {a d^{2} x^{6} - 2 \, b d x^{3} \cos \left (d x^{3} + c\right ) + 2 \, b \sin \left (d x^{3} + c\right )}{6 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 61, normalized size = 1.39 \[ \frac {\frac {{\left ({\left (d x^{3} + c\right )}^{2} - 2 \, {\left (d x^{3} + c\right )} c\right )} a}{d} - \frac {2 \, {\left (d x^{3} \cos \left (d x^{3} + c\right ) - \sin \left (d x^{3} + c\right )\right )} b}{d}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 67, normalized size = 1.52 \[ \frac {x^{6} a \,d^{2}-4 x^{3} b d \left (\cos ^{2}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )+2 x^{3} b d +4 \sin \left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right ) b \cos \left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )}{6 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 37, normalized size = 0.84 \[ \frac {1}{6} \, a x^{6} - \frac {{\left (d x^{3} \cos \left (d x^{3} + c\right ) - \sin \left (d x^{3} + c\right )\right )} b}{3 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 38, normalized size = 0.86 \[ \frac {a\,x^6}{6}+\frac {\frac {b\,\sin \left (d\,x^3+c\right )}{3}-\frac {b\,d\,x^3\,\cos \left (d\,x^3+c\right )}{3}}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.53, size = 49, normalized size = 1.11 \[ \begin {cases} \frac {a x^{6}}{6} - \frac {b x^{3} \cos {\left (c + d x^{3} \right )}}{3 d} + \frac {b \sin {\left (c + d x^{3} \right )}}{3 d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{6} \left (a + b \sin {\relax (c )}\right )}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________