Integrand size = 18, antiderivative size = 139 \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {\left (4 c^2 d+e\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {e \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}+\frac {\left (4 c^2 d+e\right ) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^3} \]
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Time = 0.24 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5909, 5881, 3384, 3379, 3382, 5887, 5556} \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^3}-\frac {e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5881
Rule 5887
Rule 5909
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{a+b \text {arccosh}(c x)}+\frac {e x^2}{a+b \text {arccosh}(c x)}\right ) \, dx \\ & = d \int \frac {1}{a+b \text {arccosh}(c x)} \, dx+e \int \frac {x^2}{a+b \text {arccosh}(c x)} \, dx \\ & = -\frac {d \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3} \\ & = -\frac {e \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3}+\frac {\left (d \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c}-\frac {\left (d \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c} \\ & = -\frac {d \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3}-\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3} \\ & = -\frac {d \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c}+\frac {\left (e \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3}+\frac {\left (e \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3}-\frac {\left (e \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3}-\frac {\left (e \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3} \\ & = -\frac {d \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {e \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {e \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c}+\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\frac {-\left (\left (4 c^2 d+e\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )\right )-e \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+4 c^2 d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b c^3} \]
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Time = 0.94 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\) | \(178\) |
default | \(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\) | \(178\) |
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\[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {e x^{2} + d}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\int \frac {d + e x^{2}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {e x^{2} + d}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {e x^{2} + d}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\int \frac {e\,x^2+d}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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