Optimal. Leaf size=86 \[ \frac{1}{2} x \left (2 a^2 C+4 a b B+b^2 C\right )+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a C+2 b B) \sin (c+d x)}{2 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.244916, antiderivative size = 86, 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, 2990, 3023, 2735, 3770} \[ \frac{1}{2} x \left (2 a^2 C+4 a b B+b^2 C\right )+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a C+2 b B) \sin (c+d x)}{2 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3029
Rule 2990
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 ^2(c+d x) \, dx &=\int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac{b C (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac{1}{2} \int \left (2 a^2 B+\left (4 a b B+2 a^2 C+b^2 C\right ) \cos (c+d x)+b (2 b B+3 a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{b (2 b B+3 a C) \sin (c+d x)}{2 d}+\frac{b C (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac{1}{2} \int \left (2 a^2 B+\left (4 a b B+2 a^2 C+b^2 C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (4 a b B+2 a^2 C+b^2 C\right ) x+\frac{b (2 b B+3 a C) \sin (c+d x)}{2 d}+\frac{b C (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\left (a^2 B\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (4 a b B+2 a^2 C+b^2 C\right ) x+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (2 b B+3 a C) \sin (c+d x)}{2 d}+\frac{b C (a+b \cos (c+d x)) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.219161, size = 120, normalized size = 1.4 \[ \frac{2 (c+d x) \left (2 a^2 C+4 a b B+b^2 C\right )-4 a^2 B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 a^2 B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 b (2 a C+b B) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 120, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{2}Cx}{2}}+{\frac{{b}^{2}Cc}{2\,d}}+{\frac{{b}^{2}B\sin \left ( dx+c \right ) }{d}}+2\,{\frac{abC\sin \left ( dx+c \right ) }{d}}+2\,abBx+2\,{\frac{Babc}{d}}+{a}^{2}Cx+{\frac{C{a}^{2}c}{d}}+{\frac{{a}^{2}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.052, size = 134, normalized size = 1.56 \begin{align*} \frac{4 \,{\left (d x + c\right )} C a^{2} + 8 \,{\left (d x + c\right )} B a b +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 2 \, B a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, C a b \sin \left (d x + c\right ) + 4 \, B b^{2} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47149, size = 213, normalized size = 2.48 \begin{align*} \frac{B a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - B a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, C a^{2} + 4 \, B a b + C b^{2}\right )} d x +{\left (C b^{2} \cos \left (d x + c\right ) + 4 \, C a b + 2 \, B b^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \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 = 1.61288, size = 240, normalized size = 2.79 \begin{align*} \frac{2 \, B a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, B a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) +{\left (2 \, C a^{2} + 4 \, B a b + C b^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B b^{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} + 4 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, B b^{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 )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]