Integrand size = 20, antiderivative size = 157 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]
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Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3399, 3394, 3384, 3379, 3382} \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 3399
Rubi steps \begin{align*} \text {integral}& = \left (4 a^2\right ) \int \frac {\sin ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right )}{(c+d x)^2} \, dx \\ & = -\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (8 i a^2 f\right ) \int \left (-\frac {i \sinh (e+f x)}{4 (c+d x)}-\frac {i \sinh (2 e+2 f x)}{8 (c+d x)}\right ) \, dx}{d} \\ & = -\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (a^2 f\right ) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{d}+\frac {\left (2 a^2 f\right ) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d} \\ & = -\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d}+\frac {\left (2 a^2 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^2 f \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d}+\frac {\left (2 a^2 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 {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 \left (-3 d-4 d \cosh (e+f x)-d \cosh (2 (e+f x))+2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )+4 f (c+d x) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+4 c f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 d f x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \]
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Time = 0.51 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.96
method | result | size |
risch | \(-\frac {f \,a^{2} {\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d^{2}}-\frac {f \,a^{2} {\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}-\frac {f \,a^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}-\frac {f \,a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}\) | \(308\) |
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Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (156) = 312\).
Time = 0.25 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.29 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^{2} d \cosh \left (f x + e\right )^{2} + a^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d \cosh \left (f x + e\right ) + 3 \, a^{2} d - 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \cosh {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\cosh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]
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Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {2}{d^{2} x + c d}\right )} - a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (156) = 312\).
Time = 0.35 (sec) , antiderivative size = 1134, normalized size of antiderivative = 7.22 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
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