Optimal. Leaf size=14 \[ -2 \sqrt{9-\sin ^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0379825, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {12, 261} \[ -2 \sqrt{9-\sin ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 261
Rubi steps
\begin{align*} \int \frac{\sin (2 x)}{\sqrt{9-\sin ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 x}{\sqrt{9-x^2}} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{9-x^2}} \, dx,x,\sin (x)\right )\\ &=-2 \sqrt{9-\sin ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0151059, size = 14, normalized size = 1. \[ -2 \sqrt{9-\sin ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.018, size = 13, normalized size = 0.9 \begin{align*} -2\,\sqrt{9- \left ( \sin \left ( x \right ) \right ) ^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.943657, size = 16, normalized size = 1.14 \begin{align*} -2 \, \sqrt{-\sin \left (x\right )^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13119, size = 31, normalized size = 2.21 \begin{align*} -2 \, \sqrt{\cos \left (x\right )^{2} + 8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.26045, size = 12, normalized size = 0.86 \begin{align*} - 2 \sqrt{9 - \sin ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16975, size = 136, normalized size = 9.71 \begin{align*} -\frac{8 \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9} - 3\right )}}{{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9}\right )}^{2} + 18 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 6 \, \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9} + 5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]