Optimal. Leaf size=11 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.00900368, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
[In] Int[Sec[a + b*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 0.611393, size = 8, normalized size = 0.73 \[ \frac{\operatorname{atanh}{\left (\sin{\left (a + b x \right )} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/cos(b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.017952, size = 68, normalized size = 6.18 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )+\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )-\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[Sec[a + b*x],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 19, normalized size = 1.7 \[{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/cos(b*x+a),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.37825, size = 35, normalized size = 3.18 \[ \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/cos(b*x + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232269, size = 38, normalized size = 3.45 \[ \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/cos(b*x + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.03065, size = 34, normalized size = 3.09 \[ \begin{cases} - \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} - 1 \right )}}{b} + \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} + 1 \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\cos{\left (a \right )}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/cos(b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216057, size = 38, normalized size = 3.45 \[ \frac{{\rm ln}\left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) -{\rm ln}\left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/cos(b*x + a),x, algorithm="giac")
[Out]