Optimal. Leaf size=31 \[ \frac{5 \log (5 \sin (c+d x)+3 \cos (c+d x))}{34 d}+\frac{3 x}{34} \]
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Rubi [A] time = 0.0402898, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3484, 3530} \[ \frac{5 \log (5 \sin (c+d x)+3 \cos (c+d x))}{34 d}+\frac{3 x}{34} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{3+5 \tan (c+d x)} \, dx &=\frac{3 x}{34}+\frac{5}{34} \int \frac{5-3 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx\\ &=\frac{3 x}{34}+\frac{5 \log (3 \cos (c+d x)+5 \sin (c+d x))}{34 d}\\ \end{align*}
Mathematica [C] time = 0.0345453, size = 65, normalized size = 2.1 \[ -\frac{\left (\frac{5}{68}+\frac{3 i}{68}\right ) \log (-\tan (c+d x)+i)}{d}-\frac{\left (\frac{5}{68}-\frac{3 i}{68}\right ) \log (\tan (c+d x)+i)}{d}+\frac{5 \log (5 \tan (c+d x)+3)}{34 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 46, normalized size = 1.5 \begin{align*} -{\frac{5\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{68\,d}}+{\frac{3\,\arctan \left ( \tan \left ( dx+c \right ) \right ) }{34\,d}}+{\frac{5\,\ln \left ( 3+5\,\tan \left ( dx+c \right ) \right ) }{34\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48665, size = 53, normalized size = 1.71 \begin{align*} \frac{6 \, d x + 6 \, c - 5 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 10 \, \log \left (5 \, \tan \left (d x + c\right ) + 3\right )}{68 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64996, size = 119, normalized size = 3.84 \begin{align*} \frac{6 \, d x + 5 \, \log \left (\frac{25 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 9}{\tan \left (d x + c\right )^{2} + 1}\right )}{68 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.52165, size = 46, normalized size = 1.48 \begin{align*} \begin{cases} \frac{3 x}{34} + \frac{5 \log{\left (\tan{\left (c + d x \right )} + \frac{3}{5} \right )}}{34 d} - \frac{5 \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{68 d} & \text{for}\: d \neq 0 \\\frac{x}{5 \tan{\left (c \right )} + 3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31001, size = 54, normalized size = 1.74 \begin{align*} \frac{6 \, d x + 6 \, c - 5 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 10 \, \log \left ({\left | 5 \, \tan \left (d x + c\right ) + 3 \right |}\right )}{68 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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