Optimal. Leaf size=17 \[ -\frac {\log (\cos (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3879, 31} \[ -\frac {\log (\cos (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\log (1+\cos (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.12 \[ -\frac {2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 19, normalized size = 1.12 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 31, normalized size = 1.82 \[ \frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 33, normalized size = 1.94 \[ \frac {\ln \left (\sec \left (d x +c \right )\right )}{a d}-\frac {\ln \left (1+\sec \left (d x +c \right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 17, normalized size = 1.00 \[ -\frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 21, normalized size = 1.24 \[ \frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.80, size = 41, normalized size = 2.41 \[ \begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} - \frac {\log {\left (\sec {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \tan {\relax (c )}}{a \sec {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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