Optimal. Leaf size=65 \[ \frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
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Rubi [A] time = 0.02, antiderivative size = 65, 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 = {2659, 206} \[ \frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2659
Rubi steps
\begin {align*} \int \frac {1}{3+5 \cos (c+d x)} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d}\\ &=-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}+\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 1.00 \[ \frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 46, normalized size = 0.71 \[ \frac {\log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 34, normalized size = 0.52 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 0.55 \[ \frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4 d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 48, normalized size = 0.74 \[ \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 17, normalized size = 0.26 \[ \frac {\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 41, normalized size = 0.63 \[ \begin {cases} - \frac {\log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{4 d} + \frac {\log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x}{5 \cos {\relax (c )} + 3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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