Integrand size = 17, antiderivative size = 73 \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a} \]
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Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6874, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a x}-\frac {b \cosh (c+d x)}{a (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a}-\frac {b \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a} \\ & = \frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a}-\frac {\left (b \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a}-\frac {\left (b \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a} \\ & = \frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\frac {\cosh (c) \text {Chi}(d x)-\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+\sinh (c) \text {Shi}(d x)-\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{a} \]
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Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.48
method | result | size |
risch | \(-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a}\) | \(108\) |
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Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.68 \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\frac {{\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, a} \]
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\[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (74) = 148\).
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.12 \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\frac {1}{2} \, d {\left (\frac {b {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a d} - \frac {2 \, \cosh \left (d x + c\right ) \log \left (x\right )}{a d} + \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}}{a d}\right )} - {\left (\frac {\log \left (b x + a\right )}{a} - \frac {\log \left (x\right )}{a}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + {\rm Ei}\left (d x\right ) e^{c} - {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )}}{2 \, a} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (a+b\,x\right )} \,d x \]
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