Optimal. Leaf size=57 \[ -\frac {4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3777, 3919, 3794} \[ -\frac {4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 3794
Rule 3919
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (c+d x))^2} \, dx &=-\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {-3 a+a \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {4 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a}\\ &=\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {4 \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 112, normalized size = 1.96 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (12 \sin \left (c+\frac {d x}{2}\right )-10 \sin \left (c+\frac {3 d x}{2}\right )+9 d x \cos \left (c+\frac {d x}{2}\right )+3 d x \cos \left (c+\frac {3 d x}{2}\right )+3 d x \cos \left (2 c+\frac {3 d x}{2}\right )-18 \sin \left (\frac {d x}{2}\right )+9 d x \cos \left (\frac {d x}{2}\right )\right )}{24 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 80, normalized size = 1.40 \[ \frac {3 \, d x \cos \left (d x + c\right )^{2} + 6 \, d x \cos \left (d x + c\right ) + 3 \, d x - {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 50, normalized size = 0.88 \[ \frac {\frac {6 \, {\left (d x + c\right )}}{a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 56, normalized size = 0.98 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a^{2} d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 72, normalized size = 1.26 \[ -\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 35, normalized size = 0.61 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,d\,x}{6\,a^2\,d} \]
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 \[ \frac {\int \frac {1}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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