Optimal. Leaf size=35 \[ x (a B+A b)+\frac {a A \sin (c+d x)}{d}+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {3996, 3770} \[ x (a B+A b)+\frac {a A \sin (c+d x)}{d}+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rule 3996
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac {a A \sin (c+d x)}{d}-\int (-A b-a B-b B \sec (c+d x)) \, dx\\ &=(A b+a B) x+\frac {a A \sin (c+d x)}{d}+(b B) \int \sec (c+d x) \, dx\\ &=(A b+a B) x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 46, normalized size = 1.31 \[ \frac {a A \sin (c) \cos (d x)}{d}+\frac {a A \cos (c) \sin (d x)}{d}+a B x+A b x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 54, normalized size = 1.54 \[ \frac {2 \, {\left (B a + A b\right )} d x + B b \log \left (\sin \left (d x + c\right ) + 1\right ) - B b \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.47, size = 79, normalized size = 2.26 \[ \frac {B b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - B b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (B a + A b\right )} {\left (d x + c\right )} + \frac {2 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.80, size = 56, normalized size = 1.60 \[ A b x +B x a +\frac {a A \sin \left (d x +c \right )}{d}+\frac {A b c}{d}+\frac {B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 58, normalized size = 1.66 \[ \frac {2 \, {\left (d x + c\right )} B a + 2 \, {\left (d x + c\right )} A b + B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.19, size = 100, normalized size = 2.86 \[ \frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________