Optimal. Leaf size=31 \[ \frac {3 \log (3 \sin (c+d x)+5 \cos (c+d x))}{34 d}+\frac {5 x}{34} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \[ \frac {3 \log (3 \sin (c+d x)+5 \cos (c+d x))}{34 d}+\frac {5 x}{34} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin {align*} \int \frac {1}{5+3 \tan (c+d x)} \, dx &=\frac {5 x}{34}+\frac {3}{34} \int \frac {3-5 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx\\ &=\frac {5 x}{34}+\frac {3 \log (5 \cos (c+d x)+3 \sin (c+d x))}{34 d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 65, normalized size = 2.10 \[ -\frac {\left (\frac {3}{68}+\frac {5 i}{68}\right ) \log (-\tan (c+d x)+i)}{d}-\frac {\left (\frac {3}{68}-\frac {5 i}{68}\right ) \log (\tan (c+d x)+i)}{d}+\frac {3 \log (3 \tan (c+d x)+5)}{34 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 46, normalized size = 1.48 \[ \frac {10 \, d x + 3 \, \log \left (\frac {9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right )}{68 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 40, normalized size = 1.29 \[ \frac {10 \, d x + 10 \, c - 3 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{68 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 46, normalized size = 1.48 \[ -\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{68 d}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{34 d}+\frac {3 \ln \left (5+3 \tan \left (d x +c \right )\right )}{34 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 39, normalized size = 1.26 \[ \frac {10 \, d x + 10 \, c - 3 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{68 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.12, size = 49, normalized size = 1.58 \[ \frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}{34\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {3}{68}-\frac {5}{68}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {3}{68}+\frac {5}{68}{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 46, normalized size = 1.48 \[ \begin {cases} \frac {5 x}{34} + \frac {3 \log {\left (\tan {\left (c + d x \right )} + \frac {5}{3} \right )}}{34 d} - \frac {3 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{68 d} & \text {for}\: d \neq 0 \\\frac {x}{3 \tan {\relax (c )} + 5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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