Integrand size = 17, antiderivative size = 150 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\frac {\cosh (c+d x)}{a (a+b x)}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a b}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a b}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2} \]
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Time = 0.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3384, 3379, 3382, 3378} \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a b}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\cosh (c+d x)}{a (a+b x)} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a (a+b x)^2}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{a} \\ & = \frac {\cosh (c+d x)}{a (a+b x)}-\frac {d \int \frac {\sinh (c+d x)}{a+b x} \, dx}{a}+\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a^2}-\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^2}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a^2}-\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^2} \\ & = \frac {\cosh (c+d x)}{a (a+b x)}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {\left (d \cosh \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 {\left (d \sinh \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 {\cosh (c+d x)}{a (a+b x)}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a b}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a b}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\frac {\frac {a \cosh (c) \cosh (d x)}{a+b x}+\cosh (c) \text {Chi}(d x)-\frac {\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \cosh \left (c-\frac {a d}{b}\right )+a d \sinh \left (c-\frac {a d}{b}\right )\right )}{b}+\frac {a \sinh (c) \sinh (d x)}{a+b x}+\sinh (c) \text {Shi}(d x)-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b}-\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{a^2} \]
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Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.69
method | result | size |
risch | \(\frac {{\mathrm e}^{-d x -c} d}{2 a \left (\left (d x +c \right ) b +d a -c b \right )}-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) d}{2 b a}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}+\frac {d \,{\mathrm e}^{d x +c}}{2 a b \left (\frac {d a}{b}+d x \right )}+\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a b}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{2}}\) | \(254\) |
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Time = 0.25 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.80 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\frac {2 \, a b \cosh \left (d x + c\right ) + {\left ({\left (b^{2} x + a b\right )} {\rm Ei}\left (d x\right ) + {\left (b^{2} x + a b\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b^{2} x + a b\right )} {\rm Ei}\left (d x\right ) - {\left (b^{2} x + a b\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} \]
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\[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x\right )^{2}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.51 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=-\frac {1}{2} \, d {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{a b} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{a b} - \frac {b {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{2} d} - \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{2} d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (x\right )}{a^{2} d} - \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}}{a^{2} d}\right )} + {\left (\frac {1}{a b x + a^{2}} - \frac {\log \left (b x + a\right )}{a^{2}} + \frac {\log \left (x\right )}{a^{2}}\right )} \cosh \left (d x + c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 1329 vs. \(2 (152) = 304\).
Time = 0.31 (sec) , antiderivative size = 1329, normalized size of antiderivative = 8.86 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x (a+b x)^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,{\left (a+b\,x\right )}^2} \,d x \]
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