Optimal. Leaf size=219 \[ -\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(22 A+18 B+3 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac{(16 A+15 B+6 C) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}+\frac{1}{2} a^4 x (2 A+8 B+13 C)+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d} \]
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Rubi [A] time = 0.713735, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3043, 2975, 2976, 2968, 3023, 2735, 3770} \[ -\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(22 A+18 B+3 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac{(16 A+15 B+6 C) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}+\frac{1}{2} a^4 x (2 A+8 B+13 C)+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2975
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^4 (a (4 A+3 B)-a (2 A-3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx}{3 a}\\ &=\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^3 \left (a^2 (16 A+15 B+6 C)-6 a^2 (2 A+B-C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{6 a}\\ &=\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^2 \left (3 a^3 (12 A+13 B+8 C)-2 a^3 (22 A+18 B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac{(22 A+18 B+3 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x)) \left (6 a^4 (12 A+13 B+8 C)-30 a^4 (2 A+B-C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac{(22 A+18 B+3 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int \left (6 a^5 (12 A+13 B+8 C)+\left (-30 a^5 (2 A+B-C)+6 a^5 (12 A+13 B+8 C)\right ) \cos (c+d x)-30 a^5 (2 A+B-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}-\frac{(22 A+18 B+3 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int \left (6 a^5 (12 A+13 B+8 C)+6 a^5 (2 A+8 B+13 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=\frac{1}{2} a^4 (2 A+8 B+13 C) x-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}-\frac{(22 A+18 B+3 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} \left (a^4 (12 A+13 B+8 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^4 (2 A+8 B+13 C) x+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}-\frac{(22 A+18 B+3 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{(16 A+15 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{6 d}+\frac{a (4 A+3 B) (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 5.69424, size = 354, normalized size = 1.62 \[ \frac{a^4 \left (6 (2 A+8 B+13 C) (c+d x)+\frac{4 (20 A+3 (4 B+C)) \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 (20 A+3 (4 B+C)) \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-6 (12 A+13 B+8 C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+6 (12 A+13 B+8 C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{-13 A-3 B}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{13 A+3 B}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2 A \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{2 A \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+12 (B+4 C) \sin (c+d x)+3 C \sin (2 (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 279, normalized size = 1.3 \begin{align*}{\frac{20\,A{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{13\,{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) }{d}}+2\,{\frac{A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+6\,{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{4}B\tan \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{13\,{a}^{4}Cx}{2}}+{\frac{13\,{a}^{4}Cc}{2\,d}}+4\,{a}^{4}Bx+4\,{\frac{B{a}^{4}c}{d}}+4\,{\frac{{a}^{4}C\sin \left ( dx+c \right ) }{d}}+A{a}^{4}x+{\frac{A{a}^{4}c}{d}}+{\frac{{a}^{4}B\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{4}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02482, size = 432, normalized size = 1.97 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 12 \,{\left (d x + c\right )} A a^{4} + 48 \,{\left (d x + c\right )} B a^{4} + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 72 \,{\left (d x + c\right )} C a^{4} - 12 \, 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 )} - 3 \, 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 )} + 36 \, 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 )} + 24 \, C a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, B a^{4} \sin \left (d x + c\right ) + 48 \, C a^{4} \sin \left (d x + c\right ) + 72 \, A a^{4} \tan \left (d x + c\right ) + 48 \, B a^{4} \tan \left (d x + c\right ) + 12 \, C a^{4} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29112, size = 487, normalized size = 2.22 \begin{align*} \frac{6 \,{\left (2 \, A + 8 \, B + 13 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{3} + 3 \,{\left (12 \, A + 13 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (12 \, A + 13 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (3 \, C a^{4} \cos \left (d x + c\right )^{4} + 6 \,{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 2 \,{\left (20 \, A + 12 \, B + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 2 \, A a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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 = 1.28613, size = 468, normalized size = 2.14 \begin{align*} \frac{3 \,{\left (2 \, A a^{4} + 8 \, B a^{4} + 13 \, C a^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (12 \, A a^{4} + 13 \, B a^{4} + 8 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (12 \, A a^{4} + 13 \, B a^{4} + 8 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{6 \,{\left (2 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, C 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 )}^{2}} - \frac{2 \,{\left (30 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 76 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 54 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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