Optimal. Leaf size=11 \[ \frac {\log \left (\sin ^{-1}(a+b x)\right )}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 11, 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, 4639} \[ \frac {\log \left (\sin ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4639
Rule 4807
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\log \left (\sin ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 11, normalized size = 1.00 \[ \frac {\log \left (\sin ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 13, normalized size = 1.18 \[ \frac {\log \left (-\arcsin \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 12, normalized size = 1.09 \[ \frac {\log \left ({\left | \arcsin \left (b x + a\right ) \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 12, normalized size = 1.09 \[ \frac {\ln \left (\arcsin \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 11, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {asin}\left (a+b\,x\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 22, normalized size = 2.00 \[ \begin {cases} \frac {\log {\left (\operatorname {asin}{\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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