Optimal. Leaf size=11 \[ \log (\sin (x))-\log (\sin (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0208168, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3258, 615} \[ \log (\sin (x))-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3258
Rule 615
Rubi steps
\begin{align*} \int \frac{\cos (x)}{\sin (x)+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x+x^2} \, dx,x,\sin (x)\right )\\ &=\log (\sin (x))-\log (1+\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0072482, size = 11, normalized size = 1. \[ \log (\sin (x))-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 12, normalized size = 1.1 \begin{align*} \ln \left ( \sin \left ( x \right ) \right ) -\ln \left ( 1+\sin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.938084, size = 15, normalized size = 1.36 \begin{align*} -\log \left (\sin \left (x\right ) + 1\right ) + \log \left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.33544, size = 47, normalized size = 4.27 \begin{align*} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.195589, size = 10, normalized size = 0.91 \begin{align*} - \log{\left (\sin{\left (x \right )} + 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05639, size = 16, normalized size = 1.45 \begin{align*} -\log \left (\sin \left (x\right ) + 1\right ) + \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]