Integrand size = 18, antiderivative size = 257 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3} \]
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Time = 0.39 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5909, 5880, 5953, 3384, 3379, 3382, 5885} \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5880
Rule 5885
Rule 5909
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{(a+b \text {arccosh}(c x))^2}+\frac {e x^2}{(a+b \text {arccosh}(c x))^2}\right ) \, dx \\ & = d \int \frac {1}{(a+b \text {arccosh}(c x))^2} \, dx+e \int \frac {x^2}{(a+b \text {arccosh}(c x))^2} \, dx \\ & = -\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {(c d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \, dx}{b}-\frac {e \text {Subst}\left (\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3} \\ & = -\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c}+\frac {e \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3} \\ & = -\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (d \cosh \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^2 c}+\frac {\left (e \cosh \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^2 c^3}+\frac {\left (3 e \cosh \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^2 c^3}-\frac {\left (d \sinh \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^2 c}-\frac {\left (e \sinh \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^2 c^3}-\frac {\left (3 e \sinh \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^2 c^3} \\ & = -\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.32 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {4 b c^2 d \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 d x \sqrt {\frac {-1+c x}{1+c x}}+4 b c^2 e x^2 \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 e x^3 \sqrt {\frac {-1+c x}{1+c x}}-\left (4 c^2 d+e\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 e (a+b \text {arccosh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 a c^2 d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+a e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+4 b c^2 d \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+b e \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 a e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b e \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b^2 c^3 (a+b \text {arccosh}(c x))} \]
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Time = 1.12 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\) | \(465\) |
default | \(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\) | \(465\) |
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\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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