Integrand size = 14, antiderivative size = 71 \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {\cosh (c+d x)}{b (a+b x)}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \]
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Time = 0.08 (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} \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cosh (c+d x)}{b (a+b x)} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (c+d x)}{b (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b} \\ & = -\frac {\cosh (c+d x)}{b (a+b x)}+\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}{b}+\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}{b} \\ & = -\frac {\cosh (c+d x)}{b (a+b x)}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\frac {-\frac {b \cosh (c+d x)}{a+b x}+d \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \sinh \left (c-\frac {a d}{b}\right )+d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^2} \]
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Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.86
method | result | size |
risch | \(-\frac {d \,{\mathrm e}^{-d x -c}}{2 b \left (d x b +d a \right )}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 b^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{2 b^{2} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 b^{2}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (74) = 148\).
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.10 \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {2 \, b \cosh \left (d x + c\right ) - {\left ({\left (b d x + a d\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (b d x + a d\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b d x + a d\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (b d x + a d\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{3} x + a b^{2}\right )}} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\text {Timed out} \]
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none
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\frac {d {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{2 \, b} - \frac {\cosh \left (d x + c\right )}{{\left (b x + a\right )} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (74) = 148\).
Time = 0.31 (sec) , antiderivative size = 615, normalized size of antiderivative = 8.66 \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - b c d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + a d^{3} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - b d^{2} e^{\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b^{2}}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} - \frac {{\left ({\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - b c d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + a d^{3} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + b d^{2} e^{\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b^{2}}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
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