Integrand size = 14, antiderivative size = 55 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=-\frac {\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}}{b n}+\frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n} \]
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Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6847, 5995, 5879, 75} \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n}-\frac {\sqrt {a+b x^n-1} \sqrt {a+b x^n+1}}{b n} \]
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Rule 75
Rule 5879
Rule 5995
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {arccosh}(a+b x) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \text {arccosh}(x) \, dx,x,a+b x^n\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x^n\right )}{b n} \\ & = -\frac {\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}}{b n}+\frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {-\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}+\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n} \]
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\[\int x^{n -1} \operatorname {arccosh}\left (a +b \,x^{n}\right )d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.76 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right ) - \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}}{b n} \]
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\[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\int x^{n - 1} \operatorname {acosh}{\left (a + b x^{n} \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {{\left (b x^{n} + a\right )} \operatorname {arcosh}\left (b x^{n} + a\right ) - \sqrt {{\left (b x^{n} + a\right )}^{2} - 1}}{b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (51) = 102\).
Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=-\frac {b {\left (\frac {a \log \left ({\left | -a b - {\left (x^{n} {\left | b \right |} - \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{n} \]
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Time = 3.59 (sec) , antiderivative size = 303, normalized size of antiderivative = 5.51 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {x^n\,\mathrm {acosh}\left (a+b\,x^n\right )}{n}-\frac {\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^3}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^3}+\frac {4\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}{b\,\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}-\frac {8\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2\,\sqrt {a-1}\,\sqrt {a+1}}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}}{n\,\left (\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^4}-\frac {2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}+1\right )}+\frac {4\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x^n-1}}{\sqrt {a+1}-\sqrt {a+b\,x^n+1}}\right )}{b\,n} \]
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