Optimal. Leaf size=120 \[ \frac{2 (36 A+C) \sin (c+d x)}{15 a^3 d}-\frac{3 A \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{3 A x}{a^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.354935, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4085, 4020, 3787, 2637, 8} \[ \frac{2 (36 A+C) \sin (c+d x)}{15 a^3 d}-\frac{3 A \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{3 A x}{a^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4085
Rule 4020
Rule 3787
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\cos (c+d x) (-a (6 A+C)+a (3 A-2 C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos (c+d x) \left (-a^2 (27 A+2 C)+2 a^2 (9 A-C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{3 A \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \cos (c+d x) \left (-2 a^3 (36 A+C)+45 a^3 A \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{3 A \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(3 A) \int 1 \, dx}{a^3}+\frac{(2 (36 A+C)) \int \cos (c+d x) \, dx}{15 a^3}\\ &=-\frac{3 A x}{a^3}+\frac{2 (36 A+C) \sin (c+d x)}{15 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{3 A \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.82001, size = 283, normalized size = 2.36 \[ -\frac{\sec \left (\frac{c}{2}\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) \left (1125 A \sin \left (c+\frac{d x}{2}\right )-1215 A \sin \left (c+\frac{3 d x}{2}\right )+225 A \sin \left (2 c+\frac{3 d x}{2}\right )-363 A \sin \left (2 c+\frac{5 d x}{2}\right )-75 A \sin \left (3 c+\frac{5 d x}{2}\right )-15 A \sin \left (3 c+\frac{7 d x}{2}\right )-15 A \sin \left (4 c+\frac{7 d x}{2}\right )+900 A d x \cos \left (c+\frac{d x}{2}\right )+450 A d x \cos \left (c+\frac{3 d x}{2}\right )+450 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+90 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+90 A d x \cos \left (3 c+\frac{5 d x}{2}\right )-1755 A \sin \left (\frac{d x}{2}\right )+900 A d x \cos \left (\frac{d x}{2}\right )+120 C \sin \left (c+\frac{d x}{2}\right )-80 C \sin \left (c+\frac{3 d x}{2}\right )+60 C \sin \left (2 c+\frac{3 d x}{2}\right )-28 C \sin \left (2 c+\frac{5 d x}{2}\right )-160 C \sin \left (\frac{d x}{2}\right )\right )}{960 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 170, normalized size = 1.4 \begin{align*}{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{C}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{17\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-6\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44244, size = 277, normalized size = 2.31 \begin{align*} \frac{3 \, A{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac{a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + \frac{C{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489513, size = 386, normalized size = 3.22 \begin{align*} -\frac{45 \, A d x \cos \left (d x + c\right )^{3} + 135 \, A d x \cos \left (d x + c\right )^{2} + 135 \, A d x \cos \left (d x + c\right ) + 45 \, A d x -{\left (15 \, A \cos \left (d x + c\right )^{3} +{\left (117 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (57 \, A + 2 \, C\right )} \cos \left (d x + c\right ) + 72 \, A + 2 \, C\right )} \sin \left (d x + c\right )}{15 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{A \cos{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \cos{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23877, size = 204, normalized size = 1.7 \begin{align*} -\frac{\frac{180 \,{\left (d x + c\right )} A}{a^{3}} - \frac{120 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac{3 \, A 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} - 30 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 10 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, 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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