Optimal. Leaf size=33 \[ -\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{2 b e} \]
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Rubi [A] time = 0.02, antiderivative size = 33, 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 = {3123, 31} \[ -\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{2 b e} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3123
Rubi steps
\begin {align*} \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{2 a+2 b x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} (d+e x)\right )\right )}{e}\\ &=-\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{2 b e}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 93, normalized size = 2.82 \[ \frac {1}{2} \left (\frac {\log \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )}{b e}-\frac {\log \left (a \sin \left (\frac {1}{2} (d+e x)\right )+a \cos \left (\frac {1}{2} (d+e x)\right )-b \sin \left (\frac {1}{2} (d+e x)\right )+b \cos \left (\frac {1}{2} (d+e x)\right )\right )}{b e}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.75, size = 54, normalized size = 1.64 \[ -\frac {\log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) - \log \left (\sin \left (e x + d\right ) + 1\right )}{4 \, b e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 82, normalized size = 2.48 \[ \frac {e^{\left (-1\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, a - 2 \, {\left | b \right |} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, a + 2 \, {\left | b \right |} \right |}}\right )}{2 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 104, normalized size = 3.15 \[ -\frac {\ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right ) a}{2 e b \left (a -b \right )}+\frac {\ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}{2 e \left (a -b \right )}+\frac {\ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{2 e b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 66, normalized size = 2.00 \[ -\frac {\frac {\log \left (-a - b - \frac {{\left (a - b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{b} - \frac {\log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{b}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.83, size = 33, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (\frac {a+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2}}{b}\right )}{b\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.69, size = 107, normalized size = 3.24 \[ \begin {cases} \frac {\tilde {\infty } x}{\cos {\relax (d )}} & \text {for}\: a = 0 \wedge b = 0 \wedge e = 0 \\- \frac {1}{a e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + a e} & \text {for}\: b = 0 \\\frac {\log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{2 b e} & \text {for}\: a = b \\\frac {x}{2 a \sin {\relax (d )} + 2 a + 2 b \cos {\relax (d )}} & \text {for}\: e = 0 \\\frac {\log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{2 b e} - \frac {\log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{2 b e} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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