Optimal. Leaf size=82 \[ \frac{(10 A+C) \sin (c+d x)}{3 a^2 d}-\frac{2 A \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 A x}{a^2}-\frac{(A+C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.217872, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{(10 A+C) \sin (c+d x)}{3 a^2 d}-\frac{2 A \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 A x}{a^2}-\frac{(A+C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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))^2} \, dx &=-\frac{(A+C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos (c+d x) (-a (4 A+C)+a (2 A-C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 A \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \cos (c+d x) \left (-a^2 (10 A+C)+6 a^2 A \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{2 A \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{(2 A) \int 1 \, dx}{a^2}+\frac{(10 A+C) \int \cos (c+d x) \, dx}{3 a^2}\\ &=-\frac{2 A x}{a^2}+\frac{(10 A+C) \sin (c+d x)}{3 a^2 d}-\frac{2 A \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 0.764618, size = 195, normalized size = 2.38 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (-30 A \sin \left (c+\frac{d x}{2}\right )+41 A \sin \left (c+\frac{3 d x}{2}\right )+9 A \sin \left (2 c+\frac{3 d x}{2}\right )+3 A \sin \left (2 c+\frac{5 d x}{2}\right )+3 A \sin \left (3 c+\frac{5 d x}{2}\right )-36 A d x \cos \left (c+\frac{d x}{2}\right )-12 A d x \cos \left (c+\frac{3 d x}{2}\right )-12 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+66 A \sin \left (\frac{d x}{2}\right )-36 A d x \cos \left (\frac{d x}{2}\right )-12 C \sin \left (c+\frac{d x}{2}\right )+8 C \sin \left (c+\frac{3 d x}{2}\right )+12 C \sin \left (\frac{d x}{2}\right )\right )}{48 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 130, normalized size = 1.6 \begin{align*} -{\frac{A}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{5\,A}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-4\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42876, size = 223, normalized size = 2.72 \begin{align*} \frac{A{\left (\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{24 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac{12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac{a^{2} \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 )}}\right )} + \frac{C{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.483188, size = 261, normalized size = 3.18 \begin{align*} -\frac{6 \, A d x \cos \left (d x + c\right )^{2} + 12 \, A d x \cos \left (d x + c\right ) + 6 \, A d x -{\left (3 \, A \cos \left (d x + c\right )^{2} + 2 \,{\left (7 \, A + C\right )} \cos \left (d x + c\right ) + 10 \, A + C\right )} \sin \left (d x + c\right )}{3 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{A \cos{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \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 ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21883, size = 154, normalized size = 1.88 \begin{align*} -\frac{\frac{12 \,{\left (d x + c\right )} A}{a^{2}} - \frac{12 \, 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^{2}} + \frac{A 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} - 15 \, A 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 )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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