Optimal. Leaf size=169 \[ \frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (6 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]
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Rubi [A] time = 0.393534, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3029, 2975, 2968, 3021, 2748, 3767, 3768, 3770} \[ \frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (6 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\int (a+a \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^6(c+d x) \, dx\\ &=\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int (a+a \cos (c+d x)) (a (6 B+5 C)+a (3 B+5 C) \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int \left (a^2 (6 B+5 C)+\left (a^2 (3 B+5 C)+a^2 (6 B+5 C)\right ) \cos (c+d x)+a^2 (3 B+5 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{a^2 (6 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{20} \int \left (4 a^2 (9 B+10 C)+5 a^2 (6 B+7 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a^2 (6 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{4} \left (a^2 (6 B+7 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{5} \left (a^2 (9 B+10 C)\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac{a^2 (6 B+7 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^2 (6 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{8} \left (a^2 (6 B+7 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^2 (9 B+10 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^2 (6 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 1.30457, size = 280, normalized size = 1.66 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (240 (6 B+7 C) \cos ^5(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-240 (B+2 C) \sin (2 c+d x)+420 B \sin (c+2 d x)+420 B \sin (3 c+2 d x)+720 B \sin (2 c+3 d x)+90 B \sin (3 c+4 d x)+90 B \sin (5 c+4 d x)+144 B \sin (4 c+5 d x)+80 (15 B+14 C) \sin (d x)+330 C \sin (c+2 d x)+330 C \sin (3 c+2 d x)+800 C \sin (2 c+3 d x)+105 C \sin (3 c+4 d x)+105 C \sin (5 c+4 d x)+160 C \sin (4 c+5 d x))\right )}{7680 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 235, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{7\,{a}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{7\,{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{6\,{a}^{2}B\tan \left ( dx+c \right ) }{5\,d}}+{\frac{{a}^{2}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{3\,{a}^{2}B \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{4\,{a}^{2}C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{2}B \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{2}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2766, size = 375, normalized size = 2.22 \begin{align*} \frac{16 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{2} + 80 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} + 160 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} - 30 \, B a^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, C a^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, C a^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72457, size = 421, normalized size = 2.49 \begin{align*} \frac{15 \,{\left (6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (9 \, B + 10 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 15 \,{\left (6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (9 \, B + 10 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 30 \,{\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) + 24 \, B a^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.69044, size = 332, normalized size = 1.96 \begin{align*} \frac{15 \,{\left (6 \, B a^{2} + 7 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (6 \, B a^{2} + 7 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (90 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 105 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 420 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 490 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 864 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 800 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 540 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 790 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 390 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 375 \, C a^{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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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