Optimal. Leaf size=278 \[ \frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{840 d}+\frac{1}{16} a^4 x (44 A+49 B+56 C)+\frac{a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]
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Rubi [A] time = 0.763355, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4086, 4017, 3996, 3787, 2635, 8, 2637} \[ \frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{840 d}+\frac{1}{16} a^4 x (44 A+49 B+56 C)+\frac{a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4017
Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (a (4 A+7 B)+a (2 A+7 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left (3 a^2 (16 A+21 B+14 C)+2 a^2 (10 A+7 B+21 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^3 (436 A+511 B+504 C)+98 a^3 (2 A+2 B+3 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^4 (988 A+1113 B+1232 C)+6 a^4 (276 A+301 B+364 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}-\frac{\int \cos ^2(c+d x) \left (-315 a^5 (44 A+49 B+56 C)-24 a^5 (454 A+504 B+581 C) \sec (c+d x)\right ) \, dx}{2520 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{1}{8} \left (a^4 (44 A+49 B+56 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{105} \left (a^4 (454 A+504 B+581 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{1}{16} \left (a^4 (44 A+49 B+56 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} a^4 (44 A+49 B+56 C) x+\frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}\\ \end{align*}
Mathematica [A] time = 1.00692, size = 204, normalized size = 0.73 \[ \frac{a^4 (105 (323 A+352 B+392 C) \sin (c+d x)+105 (124 A+127 B+128 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+11760 A c+18480 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+4060 C \sin (3 (c+d x))+840 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+23520 C d x)}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 490, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985393, size = 652, normalized size = 2.35 \begin{align*} -\frac{192 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 2688 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 140 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1792 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 35 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \,{\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} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 448 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 13440 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 840 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \, C a^{4} \sin \left (d x + c\right )}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.529753, size = 454, normalized size = 1.63 \begin{align*} \frac{105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} d x +{\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (48 \, A + 28 \, B + 7 \, 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} + 16 \,{\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, 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 = 1.32531, size = 541, normalized size = 1.95 \begin{align*} \frac{105 \,{\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4620 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5880 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 30800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 34300 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 39200 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 110936 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 135168 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 150528 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 172032 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 159656 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 58800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 73220 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 86240 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21000 \, 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 )}^{7}}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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