Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2}{d (c+d x)}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac {b^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a b f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]
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Time = 0.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3398, 3378, 3384, 3379, 3382, 3394, 12} \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2}{d (c+d x)}+\frac {2 a b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}+\frac {b^2 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {b^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)} \]
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Rule 12
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{(c+d x)^2}+\frac {2 a b \cosh (e+f x)}{(c+d x)^2}+\frac {b^2 \cosh ^2(e+f x)}{(c+d x)^2}\right ) \, dx \\ & = -\frac {a^2}{d (c+d x)}+(2 a b) \int \frac {\cosh (e+f x)}{(c+d x)^2} \, dx+b^2 \int \frac {\cosh ^2(e+f x)}{(c+d x)^2} \, dx \\ & = -\frac {a^2}{d (c+d x)}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac {(2 a b f) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d}+\frac {\left (2 i b^2 f\right ) \int -\frac {i \sinh (2 e+2 f x)}{2 (c+d x)} \, dx}{d} \\ & = -\frac {a^2}{d (c+d x)}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac {\left (b^2 f\right ) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{d}+\frac {\left (2 a b 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 (2 a b 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^2}{d (c+d x)}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac {2 a b f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {\left (b^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 (b^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 {a^2}{d (c+d x)}-\frac {2 a b \cosh (e+f x)}{d (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac {b^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 b f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^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 = 0.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {-2 a^2 d-b^2 d-4 a b d \cosh (e+f x)-b^2 d \cosh (2 (e+f x))+2 b^2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )+4 a b f (c+d x) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+4 a b c f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 a b d f x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 b^2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )}{2 d^2 (c+d x)} \]
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Time = 0.40 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.74
method | result | size |
risch | \(-\frac {f a b \,{\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}+\frac {f a b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d^{2}}-\frac {f a b \,{\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f a b \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {a^{2}}{d \left (d x +c \right )}-\frac {b^{2}}{2 \left (d x +c \right ) d}-\frac {f \,b^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}+\frac {f \,b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}-\frac {f \,b^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,b^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}\) | \(319\) |
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Time = 0.25 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {b^{2} d \cosh \left (f x + e\right )^{2} + b^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a b d \cosh \left (f x + e\right ) + {\left (2 \, a^{2} + b^{2}\right )} d - 2 \, {\left ({\left (a b d f x + a b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d f x + a b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - {\left ({\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\left ({\left (a b d f x + a b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d f x + a b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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\[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {\left (a + b \cosh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {2}{d^{2} x + c d}\right )} - a b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (186) = 372\).
Time = 0.62 (sec) , antiderivative size = 1135, normalized size of antiderivative = 6.20 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
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