Optimal. Leaf size=217 \[ -\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+44 B+36 C) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{12 d}+\frac{(7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{8 d}+a^4 x (B+4 C)+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d} \]
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Rubi [A] time = 0.742734, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3043, 2975, 2968, 3023, 2735, 3770} \[ -\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+44 B+36 C) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{12 d}+\frac{(7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{8 d}+a^4 x (B+4 C)+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2975
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 ^5(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int (a+a \cos (c+d x))^4 (4 a (A+B)-a (A-4 C) \cos (c+d x)) \sec ^4(c+d x) \, dx}{4 a}\\ &=\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int (a+a \cos (c+d x))^3 \left (3 a^2 (7 A+8 B+4 C)-a^2 (7 A+4 B-12 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{12 a}\\ &=\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int (a+a \cos (c+d x))^2 \left (2 a^3 (35 A+44 B+36 C)-a^3 (35 A+32 B-12 C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{24 a}\\ &=\frac{(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int (a+a \cos (c+d x)) \left (3 a^4 (35 A+48 B+52 C)-15 a^4 (7 A+8 B+4 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac{(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int \left (3 a^5 (35 A+48 B+52 C)+\left (-15 a^5 (7 A+8 B+4 C)+3 a^5 (35 A+48 B+52 C)\right ) \cos (c+d x)-15 a^5 (7 A+8 B+4 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{\int \left (3 a^5 (35 A+48 B+52 C)+24 a^5 (B+4 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=a^4 (B+4 C) x-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} \left (a^4 (35 A+48 B+52 C)\right ) \int \sec (c+d x) \, dx\\ &=a^4 (B+4 C) x+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac{(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [B] time = 6.19332, size = 838, normalized size = 3.86 \[ \frac{(B+4 C) (c+d x) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d}+\frac{(-35 A-48 B-52 C) (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{(35 A+48 B+52 C) (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{(\cos (c+d x) a+a)^4 \left (4 A \sin \left (\frac{1}{2} (c+d x)\right )+B \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{96 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{(\cos (c+d x) a+a)^4 \left (4 A \sin \left (\frac{1}{2} (c+d x)\right )+B \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{96 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{(\cos (c+d x) a+a)^4 \left (5 A \sin \left (\frac{1}{2} (c+d x)\right )+5 B \sin \left (\frac{1}{2} (c+d x)\right )+3 C \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{(\cos (c+d x) a+a)^4 \left (5 A \sin \left (\frac{1}{2} (c+d x)\right )+5 B \sin \left (\frac{1}{2} (c+d x)\right )+3 C \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{C (\cos (c+d x) a+a)^4 \sin (c+d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d}+\frac{(97 A+52 B+12 C) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{(-97 A-52 B-12 C) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{A (\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4}-\frac{A (\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 294, normalized size = 1.4 \begin{align*}{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{27\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{35\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{20\,{a}^{4}B\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{13\,{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{20\,A{a}^{4}\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}}+6\,{\frac{{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{4}C\tan \left ( dx+c \right ) }{d}}+4\,{a}^{4}Cx+4\,{\frac{C{a}^{4}c}{d}}+{a}^{4}Bx+{\frac{B{a}^{4}c}{d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06335, size = 562, normalized size = 2.59 \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} + 192 \,{\left (d x + c\right )} C 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 )} - 12 \, C 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 )} + 144 \, 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 )} + 48 \, C a^{4} \sin \left (d x + c\right ) + 192 \, A a^{4} \tan \left (d x + c\right ) + 288 \, B a^{4} \tan \left (d x + c\right ) + 192 \, C 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 = 2.49951, size = 493, normalized size = 2.27 \begin{align*} \frac{48 \,{\left (B + 4 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (35 \, A + 48 \, B + 52 \, C\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 + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, C a^{4} \cos \left (d x + c\right )^{4} + 32 \,{\left (5 \, A + 5 \, B + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \,{\left (27 \, A + 16 \, B + 4 \, C\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 = 1.29461, size = 458, normalized size = 2.11 \begin{align*} \frac{\frac{48 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 24 \,{\left (B a^{4} + 4 \, C a^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (35 \, A a^{4} + 48 \, B a^{4} + 52 \, C 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} + 52 \, C 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} + 84 \, C 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} - 276 \, C 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} + 300 \, C 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 ) - 108 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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