Optimal. Leaf size=111 \[ \frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{6 d}+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (7 B+5 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.381382, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3029, 2976, 2968, 3023, 2735, 3770} \[ \frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{6 d}+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (7 B+5 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3029
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\int (a+a \cos (c+d x))^3 (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} \int (a+a \cos (c+d x))^2 (3 a B+a (3 B+5 C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{1}{6} \int (a+a \cos (c+d x)) \left (6 a^2 B+15 a^2 (B+C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{1}{6} \int \left (6 a^3 B+\left (6 a^3 B+15 a^3 (B+C)\right ) \cos (c+d x)+15 a^3 (B+C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{1}{6} \int \left (6 a^3 B+3 a^3 (7 B+5 C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} a^3 (7 B+5 C) x+\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^3 B\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^3 (7 B+5 C) x+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.253484, size = 113, normalized size = 1.02 \[ \frac{a^3 \left (9 (4 B+5 C) \sin (c+d x)+3 (B+3 C) \sin (2 (c+d x))-12 B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+42 B d x+C \sin (3 (c+d x))+30 C d x\right )}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.078, size = 153, normalized size = 1.4 \begin{align*}{\frac{5\,{a}^{3}Cx}{2}}+{\frac{5\,{a}^{3}Cc}{2\,d}}+{\frac{{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{11\,{a}^{3}C\sin \left ( dx+c \right ) }{3\,d}}+{\frac{7\,{a}^{3}Bx}{2}}+{\frac{7\,{a}^{3}Bc}{2\,d}}+{\frac{3\,{a}^{3}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+3\,{\frac{{a}^{3}B\sin \left ( dx+c \right ) }{d}}+{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{3\,d}}+{\frac{{a}^{3}B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.35925, size = 200, normalized size = 1.8 \begin{align*} \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 36 \,{\left (d x + c\right )} B a^{3} - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 9 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 12 \,{\left (d x + c\right )} C a^{3} + 6 \, B a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{3} \sin \left (d x + c\right ) + 36 \, C a^{3} \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79279, size = 254, normalized size = 2.29 \begin{align*} \frac{3 \,{\left (7 \, B + 5 \, C\right )} a^{3} d x + 3 \, B a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, C a^{3} \cos \left (d x + c\right )^{2} + 3 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \,{\left (9 \, B + 11 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{6 \, 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 [A] time = 1.63942, size = 243, normalized size = 2.19 \begin{align*} \frac{6 \, B a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, B a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (7 \, B a^{3} + 5 \, C a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (15 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 36 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 33 \, C a^{3} \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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]