Optimal. Leaf size=27 \[ \frac {A \tan (c+d x)}{d}+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3021, 2735, 3770} \[ \frac {A \tan (c+d x)}{d}+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2735
Rule 3021
Rule 3770
Rubi steps
\begin {align*} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A \tan (c+d x)}{d}+\int (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=C x+\frac {A \tan (c+d x)}{d}+B \int \sec (c+d x) \, dx\\ &=C x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 27, normalized size = 1.00 \[ \frac {A \tan (c+d x)}{d}+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 71, normalized size = 2.63 \[ \frac {2 \, C d x \cos \left (d x + c\right ) + B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.54, size = 70, normalized size = 2.59 \[ \frac {{\left (d x + c\right )} C + B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 41, normalized size = 1.52 \[ C x +\frac {A \tan \left (d x +c \right )}{d}+\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 46, normalized size = 1.70 \[ \frac {2 \, {\left (d x + c\right )} C + B {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 161, normalized size = 5.96 \[ \frac {2\,B\,\mathrm {atanh}\left (\frac {64\,B^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,B^3+64\,B\,C^2}+\frac {64\,B\,C^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,B^3+64\,B\,C^2}\right )}{d}+\frac {2\,C\,\mathrm {atan}\left (\frac {64\,C^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,B^2\,C+64\,C^3}+\frac {64\,B^2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,B^2\,C+64\,C^3}\right )}{d}-\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________