Optimal. Leaf size=24 \[ x^3 (-\cos (x))+3 x^2 \sin (x)-6 \sin (x)+6 x \cos (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3296, 2637} \[ x^3 (-\cos (x))+3 x^2 \sin (x)-6 \sin (x)+6 x \cos (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 3296
Rubi steps
\begin {align*} \int x^3 \sin (x) \, dx &=-x^3 \cos (x)+3 \int x^2 \cos (x) \, dx\\ &=-x^3 \cos (x)+3 x^2 \sin (x)-6 \int x \sin (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)+3 x^2 \sin (x)-6 \int \cos (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)-6 \sin (x)+3 x^2 \sin (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 20, normalized size = 0.83 \[ 3 \left (x^2-2\right ) \sin (x)-x \left (x^2-6\right ) \cos (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 21, normalized size = 0.88 \[ -{\left (x^{3} - 6 \, x\right )} \cos \relax (x) + 3 \, {\left (x^{2} - 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.01, size = 21, normalized size = 0.88 \[ -{\left (x^{3} - 6 \, x\right )} \cos \relax (x) + 3 \, {\left (x^{2} - 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 25, normalized size = 1.04 \[ -x^{3} \cos \relax (x )+3 x^{2} \sin \relax (x )+6 x \cos \relax (x )-6 \sin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 21, normalized size = 0.88 \[ -{\left (x^{3} - 6 \, x\right )} \cos \relax (x) + 3 \, {\left (x^{2} - 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 23, normalized size = 0.96 \[ \cos \relax (x)\,\left (6\,x-x^3\right )+\sin \relax (x)\,\left (3\,x^2-6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.64, size = 26, normalized size = 1.08 \[ - x^{3} \cos {\relax (x )} + 3 x^{2} \sin {\relax (x )} + 6 x \cos {\relax (x )} - 6 \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________