Optimal. Leaf size=24 \[ \frac{3 x}{8}-\frac{1}{4} \sin ^3(x) \cos (x)-\frac{3}{8} \sin (x) \cos (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.009457, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{3 x}{8}-\frac{1}{4} \sin ^3(x) \cos (x)-\frac{3}{8} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^4(x) \, dx &=-\frac{1}{4} \cos (x) \sin ^3(x)+\frac{3}{4} \int \sin ^2(x) \, dx\\ &=-\frac{3}{8} \cos (x) \sin (x)-\frac{1}{4} \cos (x) \sin ^3(x)+\frac{3 \int 1 \, dx}{8}\\ &=\frac{3 x}{8}-\frac{3}{8} \cos (x) \sin (x)-\frac{1}{4} \cos (x) \sin ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0016437, size = 22, normalized size = 0.92 \[ \frac{3 x}{8}-\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 18, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( x \right ) }{4} \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{3\,\sin \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.919824, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) - \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.92061, size = 59, normalized size = 2.46 \begin{align*} \frac{1}{8} \,{\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{8} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.056392, size = 24, normalized size = 1. \begin{align*} \frac{3 x}{8} - \frac{\sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06319, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) - \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]