Optimal. Leaf size=7 \[ \frac{\tanh ^{-1}(\sin (x))}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0260759, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3175, 3770} \[ \frac{\tanh ^{-1}(\sin (x))}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3175
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \sec (x) \, dx}{a}\\ &=\frac{\tanh ^{-1}(\sin (x))}{a}\\ \end{align*}
Mathematica [B] time = 0.004368, size = 37, normalized size = 5.29 \[ \frac{\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 8, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.01844, size = 28, normalized size = 4. \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.92961, size = 59, normalized size = 8.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.430734, size = 19, normalized size = 2.71 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{2 a} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13484, size = 31, normalized size = 4.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]