Optimal. Leaf size=21 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a d}-\frac {x}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3888, 3770} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a d}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3888
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac {\int (-a+a \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {x}{a}+\frac {\int \sec (c+d x) \, dx}{a}\\ &=-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 60, normalized size = 2.86 \[ -\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+d x}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 35, normalized size = 1.67 \[ -\frac {2 \, d x - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 50, normalized size = 2.38 \[ -\frac {\frac {d x + c}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 59, normalized size = 2.81 \[ -\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 78, normalized size = 3.71 \[ -\frac {\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 25, normalized size = 1.19 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {x}{a} \]
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 \[ \frac {\int \frac {\tan ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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