Integrand size = 21, antiderivative size = 95 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {i a f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {i a \sinh (e+f x)}{d (c+d x)}+\frac {i a f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \]
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Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3398, 3378, 3384, 3379, 3382} \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {i a f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {i a f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {i a \sinh (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]
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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 {i a \sinh (e+f x)}{(c+d x)^2}\right ) \, dx \\ & = -\frac {a}{d (c+d x)}+(i a) \int \frac {\sinh (e+f x)}{(c+d x)^2} \, dx \\ & = -\frac {a}{d (c+d x)}-\frac {i a \sinh (e+f x)}{d (c+d x)}+\frac {(i a f) \int \frac {\cosh (e+f x)}{c+d x} \, dx}{d} \\ & = -\frac {a}{d (c+d x)}-\frac {i a \sinh (e+f x)}{d (c+d x)}+\frac {\left (i a f \cosh \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 {\left (i a f \sinh \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 {a}{d (c+d x)}+\frac {i a f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {i a \sinh (e+f x)}{d (c+d x)}+\frac {i a f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {i a \left (f (c+d x) \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-d (-i+\sinh (e+f x))+f (c+d x) \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{d^2 (c+d x)} \]
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Time = 1.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.61
method | result | size |
risch | \(-\frac {a}{d \left (d x +c \right )}+\frac {i a f \,{\mathrm e}^{-f x -e}}{2 d \left (d f x +c f \right )}-\frac {i a f \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {i a f \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {i a f \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}}\) | \(153\) |
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Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.41 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {{\left (-i \, a d e^{\left (2 \, f x + 2 \, e\right )} + i \, a d + {\left ({\left (i \, a d f x + i \, a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + {\left (i \, a d f x + i \, a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 2 \, a d\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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Timed out. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} i \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (91) = 182\).
Time = 0.28 (sec) , antiderivative size = 630, normalized size of antiderivative = 6.63 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} i \, 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. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{{\left (c+d\,x\right )}^2} \,d x \]
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