Optimal. Leaf size=38 \[ \frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2}+\frac {b \sin (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3787, 2635, 8, 2637} \[ \frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2}+\frac {b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2637
Rule 3787
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \, dx &=a \int \cos ^2(c+d x) \, dx+b \int \cos (c+d x) \, dx\\ &=\frac {b \sin (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a \int 1 \, dx\\ &=\frac {a x}{2}+\frac {b \sin (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 35, normalized size = 0.92 \[ \frac {a (2 (c+d x)+\sin (2 (c+d x)))+4 b \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 29, normalized size = 0.76 \[ \frac {a d x + {\left (a \cos \left (d x + c\right ) + 2 \, b\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 82, normalized size = 2.16 \[ \frac {{\left (d x + c\right )} a - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.62, size = 38, normalized size = 1.00 \[ \frac {a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 34, normalized size = 0.89 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a + 4 \, b \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.81, size = 31, normalized size = 0.82 \[ \frac {a\,x}{2}+\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________