Optimal. Leaf size=47 \[ \frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \text {ArcSin}\left (a+b x^n\right )}{b n} \]
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Rubi [A]
time = 0.04, antiderivative size = 47, 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, 4887,
4715, 267} \begin {gather*} \frac {\left (a+b x^n\right ) \text {ArcSin}\left (a+b x^n\right )}{b n}+\frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4887
Rule 6847
Rubi steps
\begin {align*} \int x^{-1+n} \sin ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \sin ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \sin ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \sin ^{-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 ) \sin ^{-1}\left (a+b x^n\right )}{b n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 47, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \text {ArcSin}\left (a+b x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{n -1} \arcsin \left (a +b \,x^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 39, normalized size = 0.83 \begin {gather*} \frac {{\left (b x^{n} + a\right )} \arcsin \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 [A]
time = 2.07, size = 57, normalized size = 1.21 \begin {gather*} \frac {b x^{n} \arcsin \left (b x^{n} + a\right ) + a \arcsin \left (b x^{n} + a\right ) + \sqrt {-b^{2} x^{2 \, n} - 2 \, a b x^{n} - a^{2} + 1}}{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 = 22.11, size = 76, normalized size = 1.62 \begin {gather*} \begin {cases} \log {\left (x \right )} \operatorname {asin}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \operatorname {asin}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {x^{n} \operatorname {asin}{\left (a \right )}}{n} & \text {for}\: b = 0 \\\frac {a \operatorname {asin}{\left (a + b x^{n} \right )}}{b n} + \frac {x^{n} \operatorname {asin}{\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 [A]
time = 0.40, size = 39, normalized size = 0.83 \begin {gather*} \frac {{\left (b x^{n} + a\right )} \arcsin \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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Mupad [B]
time = 0.35, size = 109, normalized size = 2.32 \begin {gather*} \frac {x^n\,\mathrm {asin}\left (a+b\,x^n\right )}{n}+\frac {\sqrt {1-b^2\,x^{2\,n}-2\,a\,b\,x^n-a^2}}{b\,n}+\frac {a\,\ln \left (\sqrt {1-b^2\,x^{2\,n}-2\,a\,b\,x^n-a^2}-\frac {a\,b+b^2\,x^n}{\sqrt {-b^2}}\right )}{n\,\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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