Optimal. Leaf size=46 \[ -\frac {\sqrt {1+\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n} \]
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Rubi [A]
time = 0.03, 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 = {6847, 5858,
5772, 267} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 5858
Rule 6847
Rubi steps
\begin {align*} \int x^{-1+n} \sinh ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \sinh ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {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 {\text {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 = 0.03, size = 41, normalized size = 0.89 \begin {gather*} \frac {-\sqrt {1+\left (a+b x^n\right )^2}+\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, 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 = 0.26, size = 39, normalized size = 0.85 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (44) = 88\).
time = 0.37, size = 152, normalized size = 3.30 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (34) = 68\).
time = 21.20, size = 76, normalized size = 1.65 \begin {gather*} \begin {cases} \log {\left (x \right )} \operatorname {asinh}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \operatorname {asinh}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {x^{n} \operatorname {asinh}{\left (a \right )}}{n} & \text {for}\: b = 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (44) = 88\).
time = 0.44, size = 113, normalized size = 2.46 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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