Optimal. Leaf size=49 \[ \frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
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Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6715, 5250, 372, 266, 63, 206} \[ \frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 372
Rule 5250
Rule 6715
Rubi steps
\begin {align*} \int x^{-1+n} \sec ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \sec ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{\left (a+b x^n\right )^2}\right )}{2 b n}\\ &=\frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ \end {align*}
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Mathematica [B] time = 0.34, size = 130, normalized size = 2.65 \[ \frac {\left (a+b x^n\right ) \sec ^{-1}\left (a+b x^n\right )}{b n}-\frac {\sqrt {\left (a+b x^n\right )^2-1} \left (\log \left (\frac {a+b x^n}{\sqrt {\left (a+b x^n\right )^2-1}}+1\right )-\log \left (1-\frac {a+b x^n}{\sqrt {\left (a+b x^n\right )^2-1}}\right )\right )}{2 b n \left (a+b x^n\right ) \sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 92, normalized size = 1.88 \[ \frac {b x^{n} \operatorname {arcsec}\left (b x^{n} + a\right ) + 2 \, a \arctan \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right ) + \log \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 75, normalized size = 1.53 \[ \frac {b {\left (\frac {2 \, {\left (b x^{n} + a\right )} \arccos \left (\frac {1}{b x^{n} + a}\right )}{b^{2}} - \frac {\log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{b^{2}}\right )}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x^{n -1} \mathrm {arcsec}\left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 66, normalized size = 1.35 \[ \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcsec}\left (b x^{n} + a\right ) - \log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 44, normalized size = 0.90 \[ -\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (a+b\,x^n\right )}^2}}}\right )-\mathrm {acos}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]
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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