Optimal. Leaf size=68 \[ \frac{\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2} (1-2 m),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\sin ^2(x)\right )}{2 m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0374064, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2577} \[ \frac{\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac{1}{2}-m} \text{Hypergeometric2F1}\left (\frac{1}{2} (1-2 m),\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+3),\sin ^2(x)\right )}{2 m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2577
Rubi steps
\begin{align*} \int \cos ^{2 m}(x) \sin ^{2 m}(x) \, dx &=\frac{\cos ^{-1+2 m}(x) \cos ^2(x)^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2} (1-2 m),\frac{1}{2} (1+2 m);\frac{1}{2} (3+2 m);\sin ^2(x)\right ) \sin ^{1+2 m}(x)}{1+2 m}\\ \end{align*}
Mathematica [A] time = 0.0662521, size = 58, normalized size = 0.85 \[ \frac{\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2}-m,m+\frac{1}{2};m+\frac{3}{2};\sin ^2(x)\right )}{2 m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.298, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( x \right ) \right ) ^{2\,m} \left ( \sin \left ( x \right ) \right ) ^{2\,m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}, x\right ) \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 \sin ^{2 m}{\left (x \right )} \cos ^{2 m}{\left (x \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 \cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]