Optimal. Leaf size=173 \[ \frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(7 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac{(35 A+32 B) \tan (c+d x) \sec (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+a^4 B x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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Rubi [A] time = 0.522871, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2975, 2968, 3021, 2735, 3770} \[ \frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(7 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac{(35 A+32 B) \tan (c+d x) \sec (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+a^4 B x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2968
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx &=\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int (a+a \cos (c+d x))^3 (a (7 A+4 B)+4 a B \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{12} \int (a+a \cos (c+d x))^2 \left (a^2 (35 A+32 B)+12 a^2 B \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{(35 A+32 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{24} \int (a+a \cos (c+d x)) \left (15 a^3 (7 A+8 B)+24 a^3 B \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{(35 A+32 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{24} \int \left (15 a^4 (7 A+8 B)+\left (24 a^4 B+15 a^4 (7 A+8 B)\right ) \cos (c+d x)+24 a^4 B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{(35 A+32 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{24} \int \left (3 a^4 (35 A+48 B)+24 a^4 B \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a^4 B x+\frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{(35 A+32 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} \left (a^4 (35 A+48 B)\right ) \int \sec (c+d x) \, dx\\ &=a^4 B x+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{(35 A+32 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(7 A+4 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 1.71883, size = 326, normalized size = 1.88 \[ \frac{a^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^4 \left (\sec (c) (105 A \sin (2 c+d x)+544 A \sin (c+2 d x)-96 A \sin (3 c+2 d x)+81 A \sin (2 c+3 d x)+81 A \sin (4 c+3 d x)+160 A \sin (3 c+4 d x)-480 A \sin (c)+105 A \sin (d x)+48 B \sin (2 c+d x)+496 B \sin (c+2 d x)-144 B \sin (3 c+2 d x)+48 B \sin (2 c+3 d x)+48 B \sin (4 c+3 d x)+160 B \sin (3 c+4 d x)+72 B d x \cos (c)+48 B d x \cos (c+2 d x)+48 B d x \cos (3 c+2 d x)+12 B d x \cos (3 c+4 d x)+12 B d x \cos (5 c+4 d x)-480 B \sin (c)+48 B \sin (d x))-24 (35 A+48 B) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{3072 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 204, normalized size = 1.2 \begin{align*}{\frac{35\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{a}^{4}Bx+{\frac{B{a}^{4}c}{d}}+{\frac{20\,A{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+6\,{\frac{{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{27\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{20\,{a}^{4}B\tan \left ( dx+c \right ) }{3\,d}}+{\frac{4\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+2\,{\frac{{a}^{4}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25626, size = 414, normalized size = 2.39 \begin{align*} \frac{64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 48 \,{\left (d x + c\right )} B a^{4} - 3 \, 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 )} - 72 \, 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 )} - 48 \, 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 )} + 24 \, A a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, 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 )} + 192 \, A a^{4} \tan \left (d x + c\right ) + 288 \, B a^{4} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43609, size = 408, normalized size = 2.36 \begin{align*} \frac{48 \, B a^{4} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (35 \, A + 48 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (35 \, A + 48 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (160 \,{\left (A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \,{\left (27 \, A + 16 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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 = 1.2737, size = 301, normalized size = 1.74 \begin{align*} \frac{24 \,{\left (d x + c\right )} B a^{4} + 3 \,{\left (35 \, A a^{4} + 48 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (35 \, A a^{4} + 48 \, 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 (105 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 385 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 424 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 279 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 216 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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