Optimal. Leaf size=35 \[ \frac {B x}{a}-\frac {(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {4072, 3919, 3794} \[ \frac {B x}{a}-\frac {(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3794
Rule 3919
Rule 4072
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=\int \frac {B+C \sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac {B x}{a}-(B-C) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac {B x}{a}-\frac {(B-C) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.14, size = 72, normalized size = 2.06 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (B d x \cos \left (c+\frac {d x}{2}\right )+2 (C-B) \sin \left (\frac {d x}{2}\right )+B d x \cos \left (\frac {d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 44, normalized size = 1.26 \[ \frac {B d x \cos \left (d x + c\right ) + B d x - {\left (B - C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 44, normalized size = 1.26 \[ \frac {\frac {{\left (d x + c\right )} B}{a} - \frac {B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.24, size = 56, normalized size = 1.60 \[ -\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.54, size = 73, normalized size = 2.09 \[ \frac {B {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + \frac {C \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.82, size = 32, normalized size = 0.91 \[ -\frac {\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a}-\frac {B\,d\,x}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________