Optimal. Leaf size=237 \[ -\frac{4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (23 A-13 B+6 C)}{2 a^3}-\frac{(13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.544866, antiderivative size = 237, 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 = {4084, 4020, 3787, 2633, 2635, 8} \[ -\frac{4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (23 A-13 B+6 C)}{2 a^3}-\frac{(13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4084
Rule 4020
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos ^3(c+d x) (a (8 A-3 B+3 C)-5 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\cos ^3(c+d x) \left (3 a^2 (21 A-11 B+6 C)-4 a^2 (13 A-8 B+3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \cos ^3(c+d x) \left (12 a^3 (34 A-19 B+9 C)-15 a^3 (23 A-13 B+6 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(23 A-13 B+6 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac{(4 (34 A-19 B+9 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac{(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(23 A-13 B+6 C) \int 1 \, dx}{2 a^3}-\frac{(4 (34 A-19 B+9 C)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac{(23 A-13 B+6 C) x}{2 a^3}+\frac{4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d}\\ \end{align*}
Mathematica [B] time = 2.64782, size = 655, normalized size = 2.76 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) \left (-600 d x (23 A-13 B+6 C) \cos \left (c+\frac{d x}{2}\right )-600 d x (23 A-13 B+6 C) \cos \left (\frac{d x}{2}\right )-11110 A \sin \left (c+\frac{d x}{2}\right )+15380 A \sin \left (c+\frac{3 d x}{2}\right )-380 A \sin \left (2 c+\frac{3 d x}{2}\right )+4777 A \sin \left (2 c+\frac{5 d x}{2}\right )+1625 A \sin \left (3 c+\frac{5 d x}{2}\right )+230 A \sin \left (3 c+\frac{7 d x}{2}\right )+230 A \sin \left (4 c+\frac{7 d x}{2}\right )-20 A \sin \left (4 c+\frac{9 d x}{2}\right )-20 A \sin \left (5 c+\frac{9 d x}{2}\right )+5 A \sin \left (5 c+\frac{11 d x}{2}\right )+5 A \sin \left (6 c+\frac{11 d x}{2}\right )-6900 A d x \cos \left (c+\frac{3 d x}{2}\right )-6900 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-1380 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-1380 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+20410 A \sin \left (\frac{d x}{2}\right )+7560 B \sin \left (c+\frac{d x}{2}\right )-9230 B \sin \left (c+\frac{3 d x}{2}\right )+930 B \sin \left (2 c+\frac{3 d x}{2}\right )-2782 B \sin \left (2 c+\frac{5 d x}{2}\right )-750 B \sin \left (3 c+\frac{5 d x}{2}\right )-105 B \sin \left (3 c+\frac{7 d x}{2}\right )-105 B \sin \left (4 c+\frac{7 d x}{2}\right )+15 B \sin \left (4 c+\frac{9 d x}{2}\right )+15 B \sin \left (5 c+\frac{9 d x}{2}\right )+3900 B d x \cos \left (c+\frac{3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+780 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac{5 d x}{2}\right )-12760 B \sin \left (\frac{d x}{2}\right )-4500 C \sin \left (c+\frac{d x}{2}\right )+4860 C \sin \left (c+\frac{3 d x}{2}\right )-900 C \sin \left (2 c+\frac{3 d x}{2}\right )+1452 C \sin \left (2 c+\frac{5 d x}{2}\right )+300 C \sin \left (3 c+\frac{5 d x}{2}\right )+60 C \sin \left (3 c+\frac{7 d x}{2}\right )+60 C \sin \left (4 c+\frac{7 d x}{2}\right )-1800 C d x \cos \left (c+\frac{3 d x}{2}\right )-1800 C d x \cos \left (2 c+\frac{3 d x}{2}\right )-360 C d x \cos \left (2 c+\frac{5 d x}{2}\right )-360 C d x \cos \left (3 c+\frac{5 d x}{2}\right )+7020 C \sin \left (\frac{d x}{2}\right )\right )}{3840 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.122, size = 542, normalized size = 2.3 \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 [B] time = 1.46487, size = 738, normalized size = 3.11 \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 = 0.519655, size = 613, normalized size = 2.59 \begin{align*} -\frac{15 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} d x -{\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \,{\left (A - B\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (19 \, A - 9 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (869 \, A - 479 \, B + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (429 \, A - 239 \, B + 114 \, C\right )} \cos \left (d x + c\right ) + 544 \, A - 304 \, B + 144 \, C\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \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 = 1.19437, size = 432, normalized size = 1.82 \begin{align*} -\frac{\frac{30 \,{\left (d x + c\right )}{\left (23 \, A - 13 \, B + 6 \, C\right )}}{a^{3}} - \frac{20 \,{\left (51 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 76 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C \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} a^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 50 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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