Optimal. Leaf size=23 \[ \frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{2 a e} \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3124, 31} \[ \frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{2 a e} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3124
Rubi steps
\begin {align*} \int \frac {1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 a+4 a x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{e}\\ &=\frac {\log \left (1+\tan \left (\frac {1}{2} (d+e x)\right )\right )}{2 a e}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 50, normalized size = 2.17 \[ \frac {\frac {\log \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )}{e}-\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )}{e}}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 31, normalized size = 1.35 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) - \log \left (\sin \left (e x + d\right ) + 1\right )}{4 \, a e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.91 \[ \frac {e^{\left (-1\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1 \right |}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 21, normalized size = 0.91 \[ \frac {\ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{2 a e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 28, normalized size = 1.22 \[ \frac {\log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{2 \, a e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.49, size = 20, normalized size = 0.87 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1\right )}{2\,a\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 36, normalized size = 1.57 \[ \begin {cases} \frac {\log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{2 a e} & \text {for}\: e \neq 0 \\\frac {x}{2 a \sin {\relax (d )} + 2 a \cos {\relax (d )} + 2 a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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