Optimal. Leaf size=47 \[ \frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6715, 6104, 5911, 260} \[ \frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5911
Rule 6104
Rule 6715
Rubi steps
\begin {align*} \int x^{-1+n} \coth ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \coth ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n}+\frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.89 \[ \frac {\log \left (1-\left (a+b x^n\right )^2\right )+2 \left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{2 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 108, normalized size = 2.30 \[ \frac {{\left (a + 1\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + 1\right ) - {\left (a - 1\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a - 1\right ) + {\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right )} \log \left (\frac {b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + 1}{b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a - 1}\right )}{2 \, b n} \]
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 \[ \int x^{n - 1} \operatorname {arcoth}\left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 118, normalized size = 2.51 \[ \frac {x^{n} \ln \left (1+a +b \,x^{n}\right )}{2 n}-\frac {x^{n} \ln \left (-1+a +b \,x^{n}\right )}{2 n}-\frac {\ln \left (x^{n}+\frac {a -1}{b}\right ) a}{2 n b}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right ) a}{2 n b}+\frac {\ln \left (x^{n}+\frac {a -1}{b}\right )}{2 n b}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right )}{2 n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 40, normalized size = 0.85 \[ \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcoth}\left (b x^{n} + a\right ) + \log \left (-{\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.51, size = 58, normalized size = 1.23 \[ \frac {\frac {\ln \left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n-1\right )}{2}+a\,\mathrm {acoth}\left (a+b\,x^n\right )}{b\,n}+\frac {x^n\,\mathrm {acoth}\left (a+b\,x^n\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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