Optimal. Leaf size=110 \[ -\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{8 \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.175024, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3817, 4020, 3787, 2635, 8, 2637} \[ -\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{8 \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3817
Rule 4020
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos ^2(c+d x) (-5 a+3 a \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{8 \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \cos ^2(c+d x) \left (-21 a^2+16 a^2 \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{8 \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{16 \int \cos (c+d x) \, dx}{3 a^2}+\frac{7 \int \cos ^2(c+d x) \, dx}{a^2}\\ &=-\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{8 \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{7 \int 1 \, dx}{2 a^2}\\ &=\frac{7 x}{2 a^2}-\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{8 \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.371987, size = 177, normalized size = 1.61 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (147 \sin \left (c+\frac{d x}{2}\right )-239 \sin \left (c+\frac{3 d x}{2}\right )-63 \sin \left (2 c+\frac{3 d x}{2}\right )-15 \sin \left (2 c+\frac{5 d x}{2}\right )-15 \sin \left (3 c+\frac{5 d x}{2}\right )+3 \sin \left (3 c+\frac{7 d x}{2}\right )+3 \sin \left (4 c+\frac{7 d x}{2}\right )+252 d x \cos \left (c+\frac{d x}{2}\right )+84 d x \cos \left (c+\frac{3 d x}{2}\right )+84 d x \cos \left (2 c+\frac{3 d x}{2}\right )-381 \sin \left (\frac{d x}{2}\right )+252 d x \cos \left (\frac{d x}{2}\right )\right )}{192 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 122, normalized size = 1.1 \begin{align*}{\frac{1}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{7}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-3\,{\frac{\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 ) ^{2}}}+7\,{\frac{\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 [A] time = 1.73415, size = 221, normalized size = 2.01 \begin{align*} -\frac{\frac{6 \,{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac{2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{21 \, \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{42 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64118, size = 257, normalized size = 2.34 \begin{align*} \frac{21 \, d x \cos \left (d x + c\right )^{2} + 42 \, d x \cos \left (d x + c\right ) + 21 \, d x +{\left (3 \, \cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )^{2} - 43 \, \cos \left (d x + c\right ) - 32\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos ^{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 = 1.361, size = 128, normalized size = 1.16 \begin{align*} \frac{\frac{21 \,{\left (d x + c\right )}}{a^{2}} - \frac{6 \,{\left (5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 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 )}^{2} a^{2}} + \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21 \, 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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