Optimal. Leaf size=12 \[ \frac{\tan ^2(x)}{2}+\log (\tan (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0390701, antiderivative size = 17, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4335, 266, 44} \[ \frac{\sec ^2(x)}{2}+\log (\sin (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4335
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{\sin (x)}{\cos ^3(x)-\cos ^5(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1-x) x^2} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{1-x}+\frac{1}{x^2}+\frac{1}{x}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=-\log (\cos (x))+\log (\sin (x))+\frac{\sec ^2(x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0150483, size = 17, normalized size = 1.42 \[ \frac{\sec ^2(x)}{2}+\log (\sin (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.024, size = 27, normalized size = 2.3 \begin{align*}{\frac{1}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}}-\ln \left ( \cos \left ( x \right ) \right ) +{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.950405, size = 35, normalized size = 2.92 \begin{align*} \frac{1}{2 \, \cos \left (x\right )^{2}} + \frac{1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right ) - 1\right ) - \log \left (\cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.31086, size = 108, normalized size = 9. \begin{align*} -\frac{\cos \left (x\right )^{2} \log \left (\cos \left (x\right )^{2}\right ) - \cos \left (x\right )^{2} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) - 1}{2 \, \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.64249, size = 29, normalized size = 2.42 \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} - \log{\left (\cos{\left (x \right )} \right )} + \frac{1}{2 \cos ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.0692, size = 42, normalized size = 3.5 \begin{align*} \frac{\cos \left (x\right )^{2} + 1}{2 \, \cos \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\cos \left (x\right )^{2}\right ) + \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]