Integrand size = 18, antiderivative size = 49 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \]
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Time = 0.05 (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} \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2} \]
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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 {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {d \int \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = -\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\frac {2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (-2 d \cosh \left (\frac {1}{2} (e+f x)\right ) \log \left (\cosh \left (\frac {1}{2} (e+f x)\right )\right )+f (c+d x) \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{a f^2 (1+\cosh (e+f x))} \]
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Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {2 \ln \left (1-\tanh \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d +\left (\left (d x +c \right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )+d x \right ) f}{a \,f^{2}}\) | \(47\) |
risch | \(\frac {2 d x}{a f}+\frac {2 d e}{a \,f^{2}}-\frac {2 \left (d x +c \right )}{a f \left (1+{\mathrm e}^{f x +e}\right )}-\frac {2 d \ln \left (1+{\mathrm e}^{f x +e}\right )}{a \,f^{2}}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.88 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\frac {2 \, {\left (d f x \cosh \left (f x + e\right ) + d f x \sinh \left (f x + e\right ) - c f - {\left (d \cosh \left (f x + e\right ) + d \sinh \left (f x + e\right ) + d\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right )\right )}}{a f^{2} \cosh \left (f x + e\right ) + a f^{2} \sinh \left (f x + e\right ) + a f^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (37) = 74\).
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.55 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\begin {cases} \frac {c \tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f} + \frac {d x \tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f} - \frac {d x}{a f} + \frac {2 d \log {\left (\tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{a f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{a \cosh {\left (e \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.45 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=2 \, d {\left (\frac {x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} + \frac {2 \, c}{{\left (a e^{\left (-f x - e\right )} + a\right )} f} \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\frac {2 \, {\left (d f x e^{\left (f x + e\right )} - d e^{\left (f x + e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - c f - d \log \left (e^{\left (f x + e\right )} + 1\right )\right )}}{a f^{2} e^{\left (f x + e\right )} + a f^{2}} \]
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Time = 1.77 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x}{a+a \cosh (e+f x)} \, dx=\frac {2\,d\,x}{a\,f}-\frac {2\,\left (c+d\,x\right )}{a\,f\,\left ({\mathrm {e}}^{e+f\,x}+1\right )}-\frac {2\,d\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e+1\right )}{a\,f^2} \]
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