Integrand size = 18, antiderivative size = 49 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \]
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Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3399, 4269, 3556} \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2} \]
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Rule 3399
Rule 3556
Rule 4269
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x) \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = \frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (2 d \cos \left (\frac {1}{2} (e+f x)\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+f (c+d x) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{a f^2 (1+\cos (e+f x))} \]
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Time = 0.98 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {-d \ln \left (\sec ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) f \left (d x +c \right )}{a \,f^{2}}\) | \(40\) |
norman | \(\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a \,f^{2}}\) | \(60\) |
risch | \(-\frac {2 i d x}{a f}-\frac {2 i d e}{a \,f^{2}}+\frac {2 i \left (d x +c \right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a \,f^{2}}\) | \(72\) |
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none
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {{\left (d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + {\left (d f x + c f\right )} \sin \left (f x + e\right )}{a f^{2} \cos \left (f x + e\right ) + a f^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\begin {cases} \frac {c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f} + \frac {d x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f} - \frac {d \log {\left (\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{a f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{a \cos {\left (e \right )} + a} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (41) = 82\).
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.27 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {\frac {{\left ({\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} + 2 \, a f \cos \left (f x + e\right ) + a f} + \frac {c \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {d e \sin \left (f x + e\right )}{a f {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (41) = 82\).
Time = 0.33 (sec) , antiderivative size = 194, normalized size of antiderivative = 3.96 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=-\frac {d f x \tan \left (\frac {1}{2} \, f x\right ) + d f x \tan \left (\frac {1}{2} \, e\right ) - d \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x\right )^{2} + \tan \left (\frac {1}{2} \, e\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + c f \tan \left (\frac {1}{2} \, f x\right ) + c f \tan \left (\frac {1}{2} \, e\right ) + d \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x\right )^{2} + \tan \left (\frac {1}{2} \, e\right )^{2} + 1}\right )}{a f^{2} \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) - a f^{2}} \]
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Time = 15.46 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x}{a+a \cos (e+f x)} \, dx=\frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{a\,f^2}+\frac {\left (c+d\,x\right )\,2{}\mathrm {i}}{a\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}-\frac {d\,x\,2{}\mathrm {i}}{a\,f} \]
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