Optimal. Leaf size=22 \[ \frac {\tan (c+d x)}{d (a+a \sec (c+d x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3879}
\begin {gather*} \frac {\tan (c+d x)}{d (a \sec (c+d x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3879
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac {\tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} \frac {\tan \left (\frac {1}{2} (c+d x)\right )}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 17, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}\) | \(17\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}\) | \(17\) |
norman | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}\) | \(17\) |
risch | \(\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 1.05 \begin {gather*} \frac {\sin \left (d x + c\right )}{a d {\left (\cos \left (d x + c\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.31, size = 22, normalized size = 1.00 \begin {gather*} \frac {\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\sec {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 16, normalized size = 0.73 \begin {gather*} \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 16, normalized size = 0.73 \begin {gather*} \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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