Optimal. Leaf size=11 \[ \log (\sin (x))-\log (\sin (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0546386, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4391, 2707, 36, 29, 31} \[ \log (\sin (x))-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4391
Rule 2707
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\csc (x)}{\sec (x)+\tan (x)} \, dx &=\int \frac{\cot (x)}{1+\sin (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin (x)\right )-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sin (x)\right )\\ &=\log (\sin (x))-\log (1+\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0216489, size = 20, normalized size = 1.82 \[ \log (\sin (x))-2 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.089, size = 8, normalized size = 0.7 \begin{align*} -\ln \left ( 1+\csc \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.11594, size = 34, normalized size = 3.09 \begin{align*} -2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.485571, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\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 = 1.11951, 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]