Optimal. Leaf size=265 \[ -\frac{a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac{a^3 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^3 (21 A+23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac{1}{16} a^3 x (21 A+23 B+26 C)+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.610614, antiderivative size = 265, 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, 2633, 2635, 8} \[ -\frac{a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac{a^3 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^3 (21 A+23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac{1}{16} a^3 x (21 A+23 B+26 C)+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4017
Rule 3996
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \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))^3 \sin (c+d x)}{7 d}+\frac{\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (a (3 A+7 B)+a (3 A+7 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (14 a^2 (3 A+4 B+3 C)+3 a^2 (9 A+7 B+14 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (129 A+147 B+154 C)+3 a^3 (87 A+91 B+112 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac{a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{\int \cos ^3(c+d x) \left (-24 a^4 (108 A+119 B+133 C)-105 a^4 (21 A+23 B+26 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{8} \left (a^3 (21 A+23 B+26 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (a^3 (108 A+119 B+133 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^3 (21 A+23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{16} \left (a^3 (21 A+23 B+26 C)\right ) \int 1 \, dx-\frac{\left (a^3 (108 A+119 B+133 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{16} a^3 (21 A+23 B+26 C) x+\frac{a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac{a^3 (21 A+23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}
Mathematica [A] time = 1.39099, size = 204, normalized size = 0.77 \[ \frac{a^3 (105 (155 A+168 B+184 C) \sin (c+d x)+105 (61 A+63 B+64 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+1155 A \sin (4 (c+d x))+399 A \sin (5 (c+d x))+105 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+3360 A c+8820 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2380 C \sin (3 (c+d x))+630 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+10920 C d x)}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.132, size = 427, normalized size = 1.6 \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 = 0.98335, size = 574, normalized size = 2.17 \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^{3} - 1344 \,{\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^{3} + 105 \,{\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^{3} - 210 \,{\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^{3} - 1344 \,{\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^{3} + 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^{3} + 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 630 \,{\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^{3} - 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^{3} + 6720 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 630 \,{\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^{3} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.579458, size = 454, normalized size = 1.71 \begin{align*} \frac{105 \,{\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} d x +{\left (240 \, A a^{3} \cos \left (d x + c\right )^{6} + 280 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (27 \, A + 21 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \,{\left (21 \, A + 23 \, B + 18 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \,{\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \,{\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \,{\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3}\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 = 1.33252, size = 541, normalized size = 2.04 \begin{align*} \frac{105 \,{\left (21 \, A a^{3} + 23 \, B a^{3} + 26 \, C a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (2205 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2415 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2730 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 14700 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 16100 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 18200 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 41601 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 45563 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 51506 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 62592 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 72576 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 77952 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 63231 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 62853 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 71246 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 25620 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 33180 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40040 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 11235 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 11025 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 10710 \, C a^{3} \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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