Optimal. Leaf size=185 \[ -\frac{(12 A-8 B+5 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \sin (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (10 A-7 B+4 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.362868, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{(12 A-8 B+5 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \sin (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (10 A-7 B+4 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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))^2} \, dx &=-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{\int \frac{\cos ^3(c+d x) (3 a (2 A-B+C)-a (4 A-4 B+C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(10 A-7 B+4 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{\int \cos ^3(c+d x) \left (3 a^2 (12 A-8 B+5 C)-3 a^2 (10 A-7 B+4 C) \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{(10 A-7 B+4 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{(10 A-7 B+4 C) \int \cos ^2(c+d x) \, dx}{a^2}+\frac{(12 A-8 B+5 C) \int \cos ^3(c+d x) \, dx}{a^2}\\ &=-\frac{(10 A-7 B+4 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{(10 A-7 B+4 C) \int 1 \, dx}{2 a^2}-\frac{(12 A-8 B+5 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac{(10 A-7 B+4 C) x}{2 a^2}+\frac{(12 A-8 B+5 C) \sin (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{(12 A-8 B+5 C) \sin ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [B] time = 1.81922, size = 473, normalized size = 2.56 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (-36 d x (10 A-7 B+4 C) \cos \left (c+\frac{d x}{2}\right )-36 d x (10 A-7 B+4 C) \cos \left (\frac{d x}{2}\right )-156 A \sin \left (c+\frac{d x}{2}\right )+342 A \sin \left (c+\frac{3 d x}{2}\right )+118 A \sin \left (2 c+\frac{3 d x}{2}\right )+30 A \sin \left (2 c+\frac{5 d x}{2}\right )+30 A \sin \left (3 c+\frac{5 d x}{2}\right )-3 A \sin \left (3 c+\frac{7 d x}{2}\right )-3 A \sin \left (4 c+\frac{7 d x}{2}\right )+A \sin \left (4 c+\frac{9 d x}{2}\right )+A \sin \left (5 c+\frac{9 d x}{2}\right )-120 A d x \cos \left (c+\frac{3 d x}{2}\right )-120 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+516 A \sin \left (\frac{d x}{2}\right )+147 B \sin \left (c+\frac{d x}{2}\right )-239 B \sin \left (c+\frac{3 d x}{2}\right )-63 B \sin \left (2 c+\frac{3 d x}{2}\right )-15 B \sin \left (2 c+\frac{5 d x}{2}\right )-15 B \sin \left (3 c+\frac{5 d x}{2}\right )+3 B \sin \left (3 c+\frac{7 d x}{2}\right )+3 B \sin \left (4 c+\frac{7 d x}{2}\right )+84 B d x \cos \left (c+\frac{3 d x}{2}\right )+84 B d x \cos \left (2 c+\frac{3 d x}{2}\right )-381 B \sin \left (\frac{d x}{2}\right )-120 C \sin \left (c+\frac{d x}{2}\right )+164 C \sin \left (c+\frac{3 d x}{2}\right )+36 C \sin \left (2 c+\frac{3 d x}{2}\right )+12 C \sin \left (2 c+\frac{5 d x}{2}\right )+12 C \sin \left (3 c+\frac{5 d x}{2}\right )-48 C d x \cos \left (c+\frac{3 d x}{2}\right )-48 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+264 C \sin \left (\frac{d x}{2}\right )\right )}{192 a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 482, normalized size = 2.6 \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 [B] time = 1.45625, size = 657, normalized size = 3.55 \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 = 0.512127, size = 437, normalized size = 2.36 \begin{align*} -\frac{3 \,{\left (10 \, A - 7 \, B + 4 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \,{\left (10 \, A - 7 \, B + 4 \, C\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (10 \, A - 7 \, B + 4 \, C\right )} d x -{\left (2 \, A \cos \left (d x + c\right )^{4} -{\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (2 \, A - B + C\right )} \cos \left (d x + c\right )^{2} +{\left (66 \, A - 43 \, B + 28 \, C\right )} \cos \left (d x + c\right ) + 48 \, A - 32 \, B + 20 \, C\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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 = 1.251, size = 359, normalized size = 1.94 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}{\left (10 \, A - 7 \, B + 4 \, C\right )}}{a^{2}} - \frac{2 \,{\left (30 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, 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} + 40 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, 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} + 18 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 9 \, 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^{2}} + \frac{A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 27 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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