Optimal. Leaf size=55 \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]
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Rubi [A] time = 0.0544168, antiderivative size = 55, 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, 5864, 5654, 74} \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 5864
Rule 5654
Rule 74
Rubi steps
\begin{align*} \int x^{-1+n} \cosh ^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \cosh ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \cosh ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,a+b x^n\right )}{b n}\\ &=-\frac{\sqrt{-1+a+b x^n} \sqrt{1+a+b x^n}}{b n}+\frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0460324, size = 50, normalized size = 0.91 \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )-\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arccosh} \left (a+b{x}^{n}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971817, size = 53, normalized size = 0.96 \begin{align*} \frac{{\left (b x^{n} + a\right )} \operatorname{arcosh}\left (b x^{n} + a\right ) - \sqrt{{\left (b x^{n} + a\right )}^{2} - 1}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19117, size = 443, normalized size = 8.05 \begin{align*} \frac{{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + \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 )}}\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 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 [B] time = 1.18431, size = 154, normalized size = 2.8 \begin{align*} -\frac{b{\left (\frac{a \log \left ({\left | -a b -{\left (x^{n}{\left | b \right |} - \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{\left | b \right |} \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt{{\left (b x^{n} + a\right )}^{2} - 1}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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