Integrand size = 18, antiderivative size = 87 \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {a f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \]
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Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378, 3384, 3379, 3382} \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=\frac {a f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {a \cosh (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{(c+d x)^2}+\frac {a \cosh (e+f x)}{(c+d x)^2}\right ) \, dx \\ & = -\frac {a}{d (c+d x)}+a \int \frac {\cosh (e+f x)}{(c+d x)^2} \, dx \\ & = -\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {(a f) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d} \\ & = -\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {\left (a f \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d}+\frac {\left (a f \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d} \\ & = -\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {a f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.78 \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=\frac {a \left (-\frac {d (1+\cosh (e+f x))}{c+d x}+f \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{d^2} \]
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Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.71
method | result | size |
risch | \(-\frac {a}{d \left (d x +c \right )}-\frac {f a \,{\mathrm e}^{-f x -e}}{2 d \left (d x f +c f \right )}+\frac {f a \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {f a \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f a \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}}\) | \(149\) |
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Time = 0.25 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.86 \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=-\frac {2 \, a d \cosh \left (f x + e\right ) + 2 \, a d - {\left ({\left (a d f x + a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a d f x + a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (a d f x + a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a d f x + a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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Timed out. \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=-\frac {1}{2} \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (90) = 180\).
Time = 0.30 (sec) , antiderivative size = 631, normalized size of antiderivative = 7.25 \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} \, a {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} - \frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \]
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Timed out. \[ \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+a\,\mathrm {cosh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
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