Optimal. Leaf size=19 \[ \frac {\sin ^{-1}(a+b x)^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4807, 4641} \[ \frac {\sin ^{-1}(a+b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 4641
Rule 4807
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^n}{\sqrt {1-a^2-2 a b x-b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^n}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sin ^{-1}(a+b x)^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 1.00 \[ \frac {\sin ^{-1}(a+b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 22, normalized size = 1.16 \[ \frac {\arcsin \left (b x + a\right )^{n} \arcsin \left (b x + a\right )}{b n + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.95, size = 19, normalized size = 1.00 \[ \frac {\arcsin \left (b x + a\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 20, normalized size = 1.05 \[ \frac {\arcsin \left (b x +a \right )^{1+n}}{b \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 37, normalized size = 1.95 \[ \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asin}\left (a+b\,x\right )\right )}{b} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asin}\left (a+b\,x\right )}^{n+1}}{b\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 60, normalized size = 3.16 \[ \begin {cases} \frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}{\relax (a )}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {x \operatorname {asin}^{n}{\relax (a )}}{\sqrt {1 - a^{2}}} & \text {for}\: b = 0 \\\frac {\log {\left (\operatorname {asin}{\left (a + b x \right )} \right )}}{b} & \text {for}\: n = -1 \\\frac {\operatorname {asin}{\left (a + b x \right )} \operatorname {asin}^{n}{\left (a + b x \right )}}{b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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