Integrand size = 14, antiderivative size = 71 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh (a+b x)}{d (c+d x)}+\frac {b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d^2}+\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2} \]
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Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384, 3379, 3382} \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\frac {b \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\cosh (a+b x)}{d (c+d x)} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (a+b x)}{d (c+d x)}+\frac {b \int \frac {\sinh (a+b x)}{c+d x} \, dx}{d} \\ & = -\frac {\cosh (a+b x)}{d (c+d x)}+\frac {\left (b \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac {\left (b \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d} \\ & = -\frac {\cosh (a+b x)}{d (c+d x)}+\frac {b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d^2}+\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\frac {-\frac {d \cosh (a+b x)}{c+d x}+b \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )+b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \]
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Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {b \,{\mathrm e}^{-b x -a}}{2 d \left (d x b +c b \right )}+\frac {b \,{\mathrm e}^{-\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {d a -c b}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{b x +a}}{2 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-d a +c b}{d}\right )}{2 d^{2}}\) | \(133\) |
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Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (71) = 142\).
Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.11 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \cosh \left (b x + a\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\text {Timed out} \]
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none
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac {\cosh \left (b x + a\right )}{{\left (d x + c\right )} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (71) = 142\).
Time = 0.28 (sec) , antiderivative size = 615, normalized size of antiderivative = 8.66 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{2} d e^{\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} + \frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - b^{2} d e^{\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]
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Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
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