Optimal. Leaf size=198 \[ \frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(8 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{20 d}+\frac{(26 A+25 B) \tan (c+d x) \sec ^2(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{30 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
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Rubi [A] time = 0.587073, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2975, 2968, 3021, 2748, 3767, 8, 3770} \[ \frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(8 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{20 d}+\frac{(26 A+25 B) \tan (c+d x) \sec ^2(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{30 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx &=\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int (a+a \cos (c+d x))^3 (a (8 A+5 B)+a (A+5 B) \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{20} \int (a+a \cos (c+d x))^2 \left (2 a^2 (26 A+25 B)+a^2 (12 A+25 B) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{60} \int (a+a \cos (c+d x)) \left (a^3 (244 A+275 B)+a^3 (88 A+125 B) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{60} \int \left (a^4 (244 A+275 B)+\left (a^4 (88 A+125 B)+a^4 (244 A+275 B)\right ) \cos (c+d x)+a^4 (88 A+125 B) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a^4 (244 A+275 B) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{120} \int \left (8 a^4 (83 A+100 B)+105 a^4 (4 A+5 B) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{a^4 (244 A+275 B) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{8} \left (7 a^4 (4 A+5 B)\right ) \int \sec (c+d x) \, dx+\frac{1}{15} \left (a^4 (83 A+100 B)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{\left (a^4 (83 A+100 B)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{a^4 (244 A+275 B) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{(26 A+25 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{(8 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.5713, size = 306, normalized size = 1.55 \[ -\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (1680 (4 A+5 B) \cos ^5(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-960 (2 A+3 B) \sin (2 c+d x)+80 (59 A+64 B) \sin (d x)+1320 A \sin (c+2 d x)+1320 A \sin (3 c+2 d x)+3200 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+420 A \sin (3 c+4 d x)+420 A \sin (5 c+4 d x)+664 A \sin (4 c+5 d x)+930 B \sin (c+2 d x)+930 B \sin (3 c+2 d x)+3520 B \sin (2 c+3 d x)-480 B \sin (4 c+3 d x)+405 B \sin (3 c+4 d x)+405 B \sin (5 c+4 d x)+800 B \sin (4 c+5 d x))\right )}{30720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 234, normalized size = 1.2 \begin{align*}{\frac{83\,A{a}^{4}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{35\,{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{7\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{20\,{a}^{4}B\tan \left ( dx+c \right ) }{3\,d}}+{\frac{34\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{27\,{a}^{4}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{4\,{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17355, size = 508, normalized size = 2.57 \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 )} A a^{4} + 480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 320 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 60 \, A a^{4}{\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 \, B a^{4}{\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 )} - 240 \, A a^{4}{\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 )} - 360 \, B a^{4}{\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 )} + 120 \, B a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{4} \tan \left (d x + c\right ) + 960 \, B a^{4} \tan \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3947, size = 431, normalized size = 2.18 \begin{align*} \frac{105 \,{\left (4 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (4 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (83 \, A + 100 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \,{\left (28 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (17 \, A + 10 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\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 = 1.30743, size = 332, normalized size = 1.68 \begin{align*} \frac{105 \,{\left (4 \, A a^{4} + 5 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \,{\left (4 \, A a^{4} + 5 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (420 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1960 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 2450 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3584 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3160 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3950 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1500 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1395 \, B a^{4} \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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