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 = 0.0552597, antiderivative size = 47, normalized size of antiderivative = 1., 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 6715
Rule 6104
Rule 5911
Rule 260
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*}
Mathematica [A] time = 0.0357095, 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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Maple [B] time = 0.119, size = 118, normalized size = 2.5 \begin{align*}{\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{a}{2\,bn}\ln \left ({x}^{n}+{\frac{a-1}{b}} \right ) }+{\frac{a}{2\,bn}\ln \left ({x}^{n}+{\frac{1+a}{b}} \right ) }+{\frac{1}{2\,bn}\ln \left ({x}^{n}+{\frac{a-1}{b}} \right ) }+{\frac{1}{2\,bn}\ln \left ({x}^{n}+{\frac{1+a}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06526, size = 54, normalized size = 1.15 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85904, size = 359, normalized size = 7.64 \begin{align*} \frac{{\left (a + 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1\right ) -{\left (a - 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1\right ) +{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1}\right )}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{n - 1} \operatorname{arcoth}\left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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