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 = 0.113096, antiderivative size = 58, normalized size of antiderivative = 1., 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 6715
Rule 6313
Rule 1961
Rule 12
Rule 203
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*}
Mathematica [A] time = 0.247847, 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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Maple [F] time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arcsech} \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 = 0.99902, size = 54, normalized size = 0.93 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.60826, size = 1138, normalized size = 19.62 \begin{align*} \frac{2 \,{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac{\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 )}} + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a}\right ) + a \log \left (\frac{\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 )}} + 1}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - a \log \left (\frac{\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 )}} - 1}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - 2 \, \arctan \left (\frac{{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\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^{2} \cosh \left (n \log \left (x\right )\right )^{2} + b^{2} \sinh \left (n \log \left (x\right )\right )^{2} + 2 \, a b \cosh \left (n \log \left (x\right )\right ) + a^{2} + 2 \,{\left (b^{2} \cosh \left (n \log \left (x\right )\right ) + a b\right )} \sinh \left (n \log \left (x\right )\right ) - 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{arsech}\left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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