Optimal. Leaf size=43 \[ \frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2}-\frac {b \cos ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3486, 2635, 8} \[ \frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2}-\frac {b \cos ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 3486
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {b \cos ^2(c+d x)}{2 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac {b \cos ^2(c+d x)}{2 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 \cos ^2(c+d x)}{2 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 46, normalized size = 1.07 \[ \frac {a (c+d x)}{2 d}+\frac {a \sin (2 (c+d x))}{4 d}-\frac {b \cos ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 35, normalized size = 0.81 \[ \frac {a d x - b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.72, size = 146, normalized size = 3.40 \[ \frac {2 \, a d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 2 \, a d x \tan \left (d x\right )^{2} + 2 \, a d x \tan \relax (c)^{2} - b \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 2 \, a \tan \left (d x\right )^{2} \tan \relax (c) - 2 \, a \tan \left (d x\right ) \tan \relax (c)^{2} + 2 \, a d x + b \tan \left (d x\right )^{2} + 4 \, b \tan \left (d x\right ) \tan \relax (c) + b \tan \relax (c)^{2} + 2 \, a \tan \left (d x\right ) + 2 \, a \tan \relax (c) - b}{4 \, {\left (d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + d \tan \left (d x\right )^{2} + d \tan \relax (c)^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 41, normalized size = 0.95 \[ \frac {-\frac {\left (\cos ^{2}\left (d x +c \right )\right ) b}{2}+a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 38, normalized size = 0.88 \[ \frac {{\left (d x + c\right )} a + \frac {a \tan \left (d x + c\right ) - b}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.68, size = 31, normalized size = 0.72 \[ \frac {a\,x}{2}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b}{2}-\frac {a\,\mathrm {tan}\left (c+d\,x\right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________