Optimal. Leaf size=58 \[ \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \tan ^{-1}\left (\sqrt {\frac {-a-b x^n+1}{a+b x^n+1}}\right )}{b n} \]
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Rubi [A] time = 0.11, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6715, 6313, 1961, 12, 203} \[ \frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \tan ^{-1}\left (\sqrt {\frac {-a-b x^n+1}{a+b x^n+1}}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 1961
Rule 6313
Rule 6715
Rubi steps
\begin {align*} \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {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 {\operatorname {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) \operatorname {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 \operatorname {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 \tan ^{-1}\left (\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 84, normalized size = 1.45 \[ \frac {\frac {\sqrt {1-\left (a+b x^n\right )^2} \sin ^{-1}\left (a+b x^n\right )}{\sqrt {-\frac {a+b x^n-1}{a+b x^n+1}} \left (a+b x^n+1\right )}+\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 385, normalized size = 6.64 \[ \frac {2 \, {\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right )} \log \left (\frac {\sqrt {-\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} + 1}{b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a}\right ) + a \log \left (\frac {\sqrt {-\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} + 1}{\cosh \left (n \log \relax (x)\right ) + \sinh \left (n \log \relax (x)\right )}\right ) - a \log \left (\frac {\sqrt {-\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} - 1}{\cosh \left (n \log \relax (x)\right ) + \sinh \left (n \log \relax (x)\right )}\right ) - 2 \, \arctan \left (\frac {{\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )} \sqrt {-\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}}{b^{2} \cosh \left (n \log \relax (x)\right )^{2} + b^{2} \sinh \left (n \log \relax (x)\right )^{2} + 2 \, a b \cosh \left (n \log \relax (x)\right ) + a^{2} + 2 \, {\left (b^{2} \cosh \left (n \log \relax (x)\right ) + a b\right )} \sinh \left (n \log \relax (x)\right ) - 1}\right )}{2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{n - 1} \operatorname {arsech}\left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int x^{-1+n} \mathrm {arcsech}\left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 40, normalized size = 0.69 \[ \frac {{\left (b x^{n} + a\right )} \operatorname {arsech}\left (b x^{n} + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} - 1}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.35, size = 54, normalized size = 0.93 \[ \frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{a+b\,x^n}-1}\,\sqrt {\frac {1}{a+b\,x^n}+1}}\right )+\mathrm {acosh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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