Optimal. Leaf size=11 \[ \log (\sin (x))-\log (\sin (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4391, 2707, 36, 29, 31} \[ \log (\sin (x))-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2707
Rule 4391
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 = 0.02, 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]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 13, normalized size = 1.18 \[ \log \left (\frac {1}{2} \, \sin \relax (x)\right ) - \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 12, normalized size = 1.09 \[ -\log \left (\sin \relax (x) + 1\right ) + \log \left ({\left | \sin \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 8, normalized size = 0.73 \[ -\ln \left (1+\csc \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.57, size = 25, normalized size = 2.27 \[ -2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.57, size = 15, normalized size = 1.36 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc {\relax (x )}}{\tan {\relax (x )} + \sec {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________