Integrand size = 14, antiderivative size = 58 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \arctan \left (\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{b n} \]
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Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6847, 6448, 1983, 12, 209} \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \arctan \left (\sqrt {\frac {-a-b x^n+1}{a+b x^n+1}}\right )}{b n} \]
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Rule 12
Rule 209
Rule 1983
Rule 6448
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {sech}^{-1}(a+b x) \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {\sqrt {\frac {1-a-b x}{1+a+b x}}}{1-a-b x} \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \arctan \left (\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{b n} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.83 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )+\frac {2 \sqrt {-\frac {-1+a+b x^n}{1+a+b x^n}} \sqrt {1-\left (a+b x^n\right )^2} \arctan \left (\frac {\sqrt {1-\left (a+b x^n\right )^2}}{-1+a+b x^n}\right )}{-1+a+b x^n}}{b n} \]
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\[\int x^{-1+n} \operatorname {arcsech}\left (a +b \,x^{n}\right )d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (54) = 108\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 6.64 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {2 \, {\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac {\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 )}} + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a}\right ) + a \log \left (\frac {\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 )}} + 1}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - a \log \left (\frac {\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 )}} - 1}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - 2 \, \arctan \left (\frac {{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\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^{2} \cosh \left (n \log \left (x\right )\right )^{2} + b^{2} \sinh \left (n \log \left (x\right )\right )^{2} + 2 \, a b \cosh \left (n \log \left (x\right )\right ) + a^{2} + 2 \, {\left (b^{2} \cosh \left (n \log \left (x\right )\right ) + a b\right )} \sinh \left (n \log \left (x\right )\right ) - 1}\right )}{2 \, b n} \]
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Timed out. \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\text {Timed out} \]
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none
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {{\left (b x^{n} + a\right )} \operatorname {arsech}\left (b x^{n} + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} - 1}\right )}{b n} \]
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\[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\int { x^{n - 1} \operatorname {arsech}\left (b x^{n} + a\right ) \,d x } \]
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Time = 5.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{a+b\,x^n}-1}\,\sqrt {\frac {1}{a+b\,x^n}+1}}\right )+\mathrm {acosh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]
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