Optimal. Leaf size=46 \[ \frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\sqrt {\left (a+b x^n\right )^2+1}}{b n} \]
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Rubi [A] time = 0.05, antiderivative size = 46, 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, 5863, 5653, 261} \[ \frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\sqrt {\left (a+b x^n\right )^2+1}}{b n} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 5863
Rule 6715
Rubi steps
\begin {align*} \int x^{-1+n} \sinh ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \sinh ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,a+b x^n\right )}{b n}\\ &=-\frac {\sqrt {1+\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 41, normalized size = 0.89 \[ \frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )-\sqrt {\left (a+b x^n\right )^2+1}}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 152, normalized size = 3.30 \[ \frac {{\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}\right ) - \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 113, normalized size = 2.46 \[ -\frac {b {\left (\frac {a \log \left (-a b - {\left (x^{n} {\left | b \right |} - \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 1}\right )} {\left | b \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt {{\left (b x^{n} + a\right )}^{2} + 1}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int x^{-1+n} \arcsinh \left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 39, normalized size = 0.85 \[ \frac {{\left (b x^{n} + a\right )} \operatorname {arsinh}\left (b x^{n} + a\right ) - \sqrt {{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 99, normalized size = 2.15 \[ \frac {x^n\,\mathrm {asinh}\left (a+b\,x^n\right )}{n}-\frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1}}{b\,n}+\frac {a\,\ln \left (\frac {a\,b+b^2\,x^n}{\sqrt {b^2}}+\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1}\right )}{n\,\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 71.05, size = 76, normalized size = 1.65 \[ \begin {cases} \log {\relax (x )} \operatorname {asinh}{\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{n} \operatorname {asinh}{\relax (a )}}{n} & \text {for}\: b = 0 \\\log {\relax (x )} \operatorname {asinh}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {a \operatorname {asinh}{\left (a + b x^{n} \right )}}{b n} + \frac {x^{n} \operatorname {asinh}{\left (a + b x^{n} \right )}}{n} - \frac {\sqrt {a^{2} + 2 a b x^{n} + b^{2} x^{2 n} + 1}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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