Optimal. Leaf size=10 \[ \sin (x)-\log (\sin (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0678942, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4391, 2833, 43} \[ \sin (x)-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4391
Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin (x)}{\sec (x)+\tan (x)} \, dx &=\int \frac{\cos (x) \sin (x)}{1+\sin (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,\sin (x)\right )\\ &=-\log (1+\sin (x))+\sin (x)\\ \end{align*}
Mathematica [A] time = 0.0206266, size = 19, normalized size = 1.9 \[ \sin (x)-2 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.089, size = 11, normalized size = 1.1 \begin{align*} -\ln \left ( \sin \left ( x \right ) +1 \right ) +\sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.57702, size = 73, normalized size = 7.3 \begin{align*} \frac{2 \, \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}} - 2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.47127, size = 36, normalized size = 3.6 \begin{align*} -\log \left (\sin \left (x\right ) + 1\right ) + \sin \left (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 \frac{\sin{\left (x \right )}}{\tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09507, size = 14, normalized size = 1.4 \begin{align*} -\log \left (\sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]