Optimal. Leaf size=25 \[ \frac {\log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{2 c e} \]
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Rubi [A] time = 0.02, antiderivative size = 25, 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} \[ \frac {\log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{2 c e} \]
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 c \sin (d+e x)} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 a+4 c x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{e}\\ &=\frac {\log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{2 c e}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 57, normalized size = 2.28 \[ \frac {1}{2} \left (\frac {\log \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )}{c e}-\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )}{c e}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 60, normalized size = 2.40 \[ \frac {\log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} + \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) - \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right )}{4 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 23, normalized size = 0.92 \[ \frac {e^{\left (-1\right )} \log \left ({\left | c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a \right |}\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 23, normalized size = 0.92 \[ \frac {\ln \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{2 c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 1.16 \[ \frac {\log \left (a + \frac {c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{2 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.82, size = 22, normalized size = 0.88 \[ \frac {\ln \left (a+c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )}{2\,c\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 63, normalized size = 2.52 \[ \begin {cases} \frac {x}{2 a \cos {\relax (d )} + 2 a} & \text {for}\: c = 0 \wedge e = 0 \\\frac {\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{2 a e} & \text {for}\: c = 0 \\\frac {x}{2 a \cos {\relax (d )} + 2 a + 2 c \sin {\relax (d )}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {a}{c} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{2 c e} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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