Optimal. Leaf size=60 \[ \frac{a^2 B \tan (c+d x)}{d}+\frac{a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac{b^2 C \sin (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.242722, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3029, 2988, 3023, 2735, 3770} \[ \frac{a^2 B \tan (c+d x)}{d}+\frac{a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac{b^2 C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3029
Rule 2988
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac{a^2 B \tan (c+d x)}{d}-\int \left (-a (2 b B+a C)-b (b B+2 a C) \cos (c+d x)-b^2 C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{b^2 C \sin (c+d x)}{d}+\frac{a^2 B \tan (c+d x)}{d}-\int (-a (2 b B+a C)-b (b B+2 a C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=b (b B+2 a C) x+\frac{b^2 C \sin (c+d x)}{d}+\frac{a^2 B \tan (c+d x)}{d}+(a (2 b B+a C)) \int \sec (c+d x) \, dx\\ &=b (b B+2 a C) x+\frac{a (2 b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 C \sin (c+d x)}{d}+\frac{a^2 B \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.473368, size = 109, normalized size = 1.82 \[ \frac{a^2 B \tan (c+d x)+b (c+d x) (2 a C+b B)-a (a C+2 b B) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+a (a C+2 b B) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+b^2 C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 104, normalized size = 1.7 \begin{align*}{b}^{2}Bx+2\,abCx+2\,{\frac{abB\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}B\tan \left ( dx+c \right ) }{d}}+{\frac{B{b}^{2}c}{d}}+{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}C\sin \left ( dx+c \right ) }{d}}+2\,{\frac{Cabc}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10479, size = 139, normalized size = 2.32 \begin{align*} \frac{4 \,{\left (d x + c\right )} C a b + 2 \,{\left (d x + c\right )} B b^{2} + C a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C b^{2} \sin \left (d x + c\right ) + 2 \, B a^{2} \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 = 1.59998, size = 294, normalized size = 4.9 \begin{align*} \frac{2 \,{\left (2 \, C a b + B b^{2}\right )} d x \cos \left (d x + c\right ) +{\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C b^{2} \cos \left (d x + c\right ) + B a^{2}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.69251, size = 205, normalized size = 3.42 \begin{align*} \frac{{\left (2 \, C a b + B b^{2}\right )}{\left (d x + c\right )} +{\left (C a^{2} + 2 \, B a b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (C a^{2} + 2 \, B a b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C b^{2} \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]