Integrand size = 12, antiderivative size = 98 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d} \]
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Time = 0.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5995, 5880, 5953, 3384, 3379, 3382} \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b d (a+b \text {arccosh}(c+d x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5880
Rule 5953
Rule 5995
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \text {arccosh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d} \\ & = -\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d} \\ & = -\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {-\frac {b \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \text {arccosh}(c+d x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )}{b^2 d} \]
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Time = 0.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\) | \(139\) |
default | \(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\) | \(139\) |
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\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]
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