Optimal. Leaf size=75 \[ \frac{a \log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{4 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))} \]
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Rubi [A] time = 0.0531722, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3129, 12, 3121, 31} \[ \frac{a \log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{4 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))} \]
Antiderivative was successfully verified.
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Rule 3129
Rule 12
Rule 3121
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx &=-\frac{c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac{\int -\frac{2 a}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{4 c^2}\\ &=-\frac{c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}-\frac{a \int \frac{1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{2 c^2}\\ &=-\frac{c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac{a \operatorname{Subst}\left (\int \frac{1}{2 a+2 c x} \, dx,x,\cot \left (\frac{1}{2} (d+e x)\right )\right )}{2 c^2 e}\\ &=\frac{a \log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac{c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}\\ \end{align*}
Mathematica [B] time = 0.413908, size = 229, normalized size = 3.05 \[ -\frac{\sin \left (\frac{1}{2} (d+e x)\right ) \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right ) \left (\cos (d+e x) \left (2 a^2 \log \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )-2 a^2 \log \left (\sin \left (\frac{1}{2} (d+e x)\right )\right )+a^2+2 c^2\right )+a \left (a \left (-2 \log \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )+2 \log \left (\sin \left (\frac{1}{2} (d+e x)\right )\right )-1\right )+c \sin (d+e x) \left (-2 \log \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )+2 \log \left (\sin \left (\frac{1}{2} (d+e x)\right )\right )+1\right )\right )\right )}{4 c^3 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 110, normalized size = 1.5 \begin{align*} -{\frac{a}{8\,{c}^{2}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{1}{8\,ae} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}+{\frac{a}{4\,{c}^{3}e}\ln \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{1}{8\,{c}^{2}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{a}{4\,{c}^{3}e}\ln \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01679, size = 185, normalized size = 2.47 \begin{align*} -\frac{\frac{a c + \frac{{\left (2 \, a^{2} + c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{\frac{a c^{3} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{a^{2} c^{2} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}} - \frac{2 \, a \log \left (c + \frac{a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{3}} + \frac{2 \, a \log \left (\frac{\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{3}}}{8 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34212, size = 398, normalized size = 5.31 \begin{align*} \frac{2 \, c^{2} \cos \left (e x + d\right ) + 2 \, a c \sin \left (e x + d\right ) +{\left (a^{2} \cos \left (e x + d\right ) - a c \sin \left (e x + d\right ) - a^{2}\right )} \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 ) -{\left (a^{2} \cos \left (e x + d\right ) - a c \sin \left (e x + d\right ) - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (e x + d\right ) + \frac{1}{2}\right )}{8 \,{\left (a c^{3} e \cos \left (e x + d\right ) - c^{4} e \sin \left (e x + d\right ) - a c^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15568, size = 155, normalized size = 2.07 \begin{align*} \frac{1}{8} \,{\left (\frac{2 \, a \log \left ({\left | a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + c \right |}\right )}{c^{3}} - \frac{2 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) \right |}\right )}{c^{3}} - \frac{2 \, a^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a c}{{\left (a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )\right )} a c^{2}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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