Optimal. Leaf size=151 \[ \frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac{(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac{1}{8} a^4 x (48 A+35 B)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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Rubi [A] time = 0.412612, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2976, 2968, 3023, 2735, 3770} \[ \frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac{(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac{1}{8} a^4 x (48 A+35 B)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx &=\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{4} \int (a+a \cos (c+d x))^3 (4 a A+a (4 A+7 B) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{1}{12} \int (a+a \cos (c+d x))^2 \left (12 a^2 A+a^2 (32 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac{1}{24} \int (a+a \cos (c+d x)) \left (24 a^3 A+15 a^3 (8 A+7 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac{1}{24} \int \left (24 a^4 A+\left (24 a^4 A+15 a^4 (8 A+7 B)\right ) \cos (c+d x)+15 a^4 (8 A+7 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac{1}{24} \int \left (24 a^4 A+3 a^4 (48 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{8} a^4 (48 A+35 B) x+\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{8} a^4 (48 A+35 B) x+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac{(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end{align*}
Mathematica [A] time = 0.372112, size = 138, normalized size = 0.91 \[ \frac{a^4 \left (24 (27 A+28 B) \sin (c+d x)+24 (4 A+7 B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))-96 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+96 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+576 A d x+32 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+420 B d x\right )}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 199, normalized size = 1.3 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}+{\frac{20\,A{a}^{4}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{4}B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{27\,{a}^{4}B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{35\,{a}^{4}Bx}{8}}+{\frac{35\,{a}^{4}Bc}{8\,d}}+2\,{\frac{A{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+6\,A{a}^{4}x+6\,{\frac{A{a}^{4}c}{d}}+{\frac{4\,B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}+{\frac{20\,{a}^{4}B\sin \left ( dx+c \right ) }{3\,d}}+{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01195, size = 267, normalized size = 1.77 \begin{align*} -\frac{32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 96 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 384 \,{\left (d x + c\right )} A a^{4} + 128 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 144 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 96 \,{\left (d x + c\right )} B a^{4} - 96 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 576 \, A a^{4} \sin \left (d x + c\right ) - 384 \, B a^{4} \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81413, size = 306, normalized size = 2.03 \begin{align*} \frac{3 \,{\left (48 \, A + 35 \, B\right )} a^{4} d x + 12 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (6 \, B a^{4} \cos \left (d x + c\right )^{3} + 8 \,{\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \,{\left (16 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right ) + 160 \,{\left (A + B\right )} a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \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 = 1.2994, size = 289, normalized size = 1.91 \begin{align*} \frac{24 \, A a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, A a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (48 \, A a^{4} + 35 \, B a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (120 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 105 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 424 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 385 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 520 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 511 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 216 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 279 \, 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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