Optimal. Leaf size=46 \[ a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.116204, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4076, 4047, 8, 4045, 3770} \[ a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4076
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{a C \tan (c+d x)}{d}+\int \cos (c+d x) \left (a A+a (A+B) \sec (c+d x)+a (B+C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a C \tan (c+d x)}{d}+(a (A+B)) \int 1 \, dx+\int \cos (c+d x) \left (a A+a (B+C) \sec ^2(c+d x)\right ) \, dx\\ &=a (A+B) x+\frac{a A \sin (c+d x)}{d}+\frac{a C \tan (c+d x)}{d}+(a (B+C)) \int \sec (c+d x) \, dx\\ &=a (A+B) x+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x)}{d}+\frac{a C \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0421884, size = 71, normalized size = 1.54 \[ \frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.076, size = 88, normalized size = 1.9 \begin{align*} aAx+aBx+{\frac{Aa\sin \left ( dx+c \right ) }{d}}+{\frac{Aac}{d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bac}{d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{aC\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.947619, size = 124, normalized size = 2.7 \begin{align*} \frac{2 \,{\left (d x + c\right )} A a + 2 \,{\left (d x + c\right )} B a + B a{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + C a{\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 \, C a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.523306, size = 257, normalized size = 5.59 \begin{align*} \frac{2 \,{\left (A + B\right )} a d x \cos \left (d x + c\right ) +{\left (B + C\right )} a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B + C\right )} a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (A a \cos \left (d x + c\right ) + C a\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int A \cos{\left (c + d x \right )}\, dx + \int A \cos{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int B \cos{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int B \cos{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18872, size = 181, normalized size = 3.93 \begin{align*} \frac{{\left (A a + B a\right )}{\left (d x + c\right )} +{\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]