Optimal. Leaf size=27 \[ \frac {B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {B x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 2635, 8} \[ \frac {B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {B x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 21
Rule 2635
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx &=B \int \cos ^2(c+d x) \, dx\\ &=\frac {B \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} B \int 1 \, dx\\ &=\frac {B x}{2}+\frac {B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 24, normalized size = 0.89 \[ \frac {B (2 (c+d x)+\sin (2 (c+d x)))}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 24, normalized size = 0.89 \[ \frac {B d x + B \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 33, normalized size = 1.22 \[ \frac {{\left (d x + c\right )} B + \frac {B \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 28, normalized size = 1.04 \[ \frac {B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.87, size = 50, normalized size = 1.85 \[ \frac {B\,x}{2}+\frac {B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-B\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.89, size = 68, normalized size = 2.52 \[ \begin {cases} \frac {B x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\relax (c )}\right ) \cos ^{2}{\relax (c )}}{a + b \cos {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________