Optimal. Leaf size=17 \[ a x-\frac {b \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3086, 3475} \[ a x-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3086
Rule 3475
Rubi steps
\begin {align*} \int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int (a+b \tan (c+d x)) \, dx\\ &=a x+b \int \tan (c+d x) \, dx\\ &=a x-\frac {b \log (\cos (c+d x))}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ a x-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 21, normalized size = 1.24 \[ \frac {a d x - b \log \left (-\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 27, normalized size = 1.59 \[ \frac {2 \, {\left (d x + c\right )} a + b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.34, size = 24, normalized size = 1.41 \[ a x -\frac {b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {c a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 30, normalized size = 1.76 \[ \frac {2 \, {\left (d x + c\right )} a - b \log \left (-\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.57, size = 70, normalized size = 4.12 \[ \frac {b\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {2\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {b\,\ln \left (\frac {\cos \left (c+d\,x\right )}{\cos \left (c+d\,x\right )+1}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________