Optimal. Leaf size=287 \[ \frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{840 d}+\frac{a (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]
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Rubi [A] time = 0.868061, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.22, Rules used = {3043, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{840 d}+\frac{a (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]
Antiderivative was successfully verified.
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Rule 3043
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 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int (a+a \cos (c+d x))^4 (a (4 A+7 B)+a (2 A+7 C) \cos (c+d x)) \sec ^7(c+d x) \, dx}{7 a}\\ &=\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int (a+a \cos (c+d x))^3 \left (3 a^2 (16 A+21 B+14 C)+2 a^2 (10 A+7 B+21 C) \cos (c+d x)\right ) \sec ^6(c+d x) \, dx}{42 a}\\ &=\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int (a+a \cos (c+d x))^2 \left (a^3 (436 A+511 B+504 C)+98 a^3 (2 A+2 B+3 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{210 a}\\ &=\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int (a+a \cos (c+d x)) \left (3 a^4 (988 A+1113 B+1232 C)+6 a^4 (276 A+301 B+364 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{840 a}\\ &=\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int \left (3 a^5 (988 A+1113 B+1232 C)+\left (6 a^5 (276 A+301 B+364 C)+3 a^5 (988 A+1113 B+1232 C)\right ) \cos (c+d x)+6 a^5 (276 A+301 B+364 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{840 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \sec ^2(c+d x) \tan (c+d x)}{840 d}+\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{\int \left (315 a^5 (44 A+49 B+56 C)+24 a^5 (454 A+504 B+581 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{2520 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \sec ^2(c+d x) \tan (c+d x)}{840 d}+\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{8} \left (a^4 (44 A+49 B+56 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{105} \left (a^4 (454 A+504 B+581 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^4 (44 A+49 B+56 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \sec ^2(c+d x) \tan (c+d x)}{840 d}+\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{16} \left (a^4 (44 A+49 B+56 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^4 (454 A+504 B+581 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 d}\\ &=\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \sec ^2(c+d x) \tan (c+d x)}{840 d}+\frac{(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 3.44124, size = 298, normalized size = 1.04 \[ -\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (3360 (44 A+49 B+56 C) \cos ^7(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-2 \sin (c+d x) (70 (1444 A+1291 B+1128 C) \cos (c+d x)+8 (12746 A+12936 B+12859 C) \cos (2 (c+d x))+35420 A \cos (3 (c+d x))+29056 A \cos (4 (c+d x))+4620 A \cos (5 (c+d x))+3632 A \cos (6 (c+d x))+80384 A+37205 B \cos (3 (c+d x))+32256 B \cos (4 (c+d x))+5145 B \cos (5 (c+d x))+4032 B \cos (6 (c+d x))+75264 B+36120 C \cos (3 (c+d x))+35504 C \cos (4 (c+d x))+5880 C \cos (5 (c+d x))+4648 C \cos (6 (c+d x))+72016 C)\right )}{860160 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 454, normalized size = 1.6 \begin{align*}{\frac{454\,A{a}^{4}\tan \left ( dx+c \right ) }{105\,d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7\,d}}+{\frac{48\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{227\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{105\,d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{34\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{24\,{a}^{4}B\tan \left ( dx+c \right ) }{5\,d}}+{\frac{4\,{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{12\,{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{5\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{41\,{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{49\,{a}^{4}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{2\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{3\,d}}+{\frac{11\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{6\,d}}+{\frac{11\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{4\,d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{7\,{a}^{4}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{49\,{a}^{4}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{11\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{7\,{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{83\,{a}^{4}C\tan \left ( dx+c \right ) }{15\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0551, size = 987, normalized size = 3.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09729, size = 617, normalized size = 2.15 \begin{align*} \frac{105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 16 \,{\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (44 \, A + 41 \, B + 24 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 48 \,{\left (48 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 240 \, A a^{4}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \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 = 1.28688, size = 598, normalized size = 2.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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