Optimal. Leaf size=229 \[ \frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 d}+\frac{(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]
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Rubi [A] time = 0.649648, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2975, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 d}+\frac{(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx &=\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{6} \int (a+a \cos (c+d x))^3 (3 a (3 A+2 B)+2 a (A+3 B) \cos (c+d x)) \sec ^6(c+d x) \, dx\\ &=\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{30} \int (a+a \cos (c+d x))^2 \left (a^2 (73 A+72 B)+14 a^2 (2 A+3 B) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{120} \int (a+a \cos (c+d x)) \left (3 a^3 (159 A+176 B)+6 a^3 (43 A+52 B) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{120} \int \left (3 a^4 (159 A+176 B)+\left (6 a^4 (43 A+52 B)+3 a^4 (159 A+176 B)\right ) \cos (c+d x)+6 a^4 (43 A+52 B) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{360} \int \left (315 a^4 (7 A+8 B)+24 a^4 (72 A+83 B) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{8} \left (7 a^4 (7 A+8 B)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{15} \left (a^4 (72 A+83 B)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{16} \left (7 a^4 (7 A+8 B)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^4 (72 A+83 B)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 2.07869, size = 358, normalized size = 1.56 \[ -\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (3360 (7 A+8 B) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-160 (72 A+83 B) \sin (c)+30 (125 A+88 B) \sin (d x)+3750 A \sin (2 c+d x)+15360 A \sin (c+2 d x)-1920 A \sin (3 c+2 d x)+3845 A \sin (2 c+3 d x)+3845 A \sin (4 c+3 d x)+6912 A \sin (3 c+4 d x)+735 A \sin (4 c+5 d x)+735 A \sin (6 c+5 d x)+1152 A \sin (5 c+6 d x)+2640 B \sin (2 c+d x)+15840 B \sin (c+2 d x)-4080 B \sin (3 c+2 d x)+3480 B \sin (2 c+3 d x)+3480 B \sin (4 c+3 d x)+7728 B \sin (3 c+4 d x)-240 B \sin (5 c+4 d x)+840 B \sin (4 c+5 d x)+840 B \sin (6 c+5 d x)+1328 B \sin (5 c+6 d x))\right )}{122880 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 280, normalized size = 1.2 \begin{align*}{\frac{49\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{49\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{83\,{a}^{4}B\tan \left ( dx+c \right ) }{15\,d}}+{\frac{24\,A{a}^{4}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{12\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{5\,d}}+{\frac{7\,{a}^{4}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{41\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{34\,{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{4\,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}}{d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0665, size = 626, normalized size = 2.73 \begin{align*} \frac{128 \,{\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} + 640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \,{\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^{4} + 960 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 5 \, A a^{4}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, 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 )} - 120 \, 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 )} - 120 \, 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 )} - 480 \, 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 )} + 480 \, B a^{4} \tan \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41561, size = 481, normalized size = 2.1 \begin{align*} \frac{105 \,{\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (72 \, A + 83 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 105 \,{\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \,{\left (18 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \,{\left (41 \, A + 24 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 = 1.31649, size = 378, normalized size = 1.65 \begin{align*} \frac{105 \,{\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \,{\left (7 \, A a^{4} + 8 \, 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 (735 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3000 \, 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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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