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 = 0.18, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 8
Rule 2635
Rule 2637
Rule 3787
Rule 3817
Rule 4020
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*}
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Mathematica [A] time = 0.39, 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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fricas [A] time = 0.95, size = 99, normalized size = 0.90 \[ \frac {21 \, d x \cos \left (d x + c\right )^{2} + 42 \, d x \cos \left (d x + c\right ) + 21 \, d x + {\left (3 \, \cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )^{2} - 43 \, \cos \left (d x + c\right ) - 32\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 95, normalized size = 0.86 \[ \frac {\frac {21 \, {\left (d x + c\right )}}{a^{2}} - \frac {6 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 122, normalized size = 1.11 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a^{2} d}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d}-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 164, normalized size = 1.49 \[ -\frac {\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 113, normalized size = 1.03 \[ \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-22\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{6\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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