Optimal. Leaf size=25 \[ \frac {a \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3879, 43} \[ \frac {a \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \tan (c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a+a x}{x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^2}+\frac {a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \log (\cos (c+d x))}{d}+\frac {a \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {a \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 34, normalized size = 1.36 \[ -\frac {a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - a}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 106, normalized size = 4.24 \[ \frac {a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {3 \, a + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 25, normalized size = 1.00 \[ \frac {a \sec \left (d x +c \right )}{d}+\frac {a \ln \left (\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 26, normalized size = 1.04 \[ -\frac {a \log \left (\cos \left (d x + c\right )\right ) - \frac {a}{\cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 40, normalized size = 1.60 \[ \frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 37, normalized size = 1.48 \[ \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right ) \tan {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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