Optimal. Leaf size=47 \[ \frac {(a+b x) \sin ^{-1}(a+b x)^2}{b}+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}-2 x \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4803, 4619, 4677, 8} \[ \frac {(a+b x) \sin ^{-1}(a+b x)^2}{b}+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}-2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4619
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \sin ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{b}\\ &=-2 x+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.04 \[ \frac {-2 (a+b x)+(a+b x) \sin ^{-1}(a+b x)^2+2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 53, normalized size = 1.13 \[ \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2} - 2 \, b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 52, normalized size = 1.11 \[ \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{b} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )}{b} - \frac {2 \, {\left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 48, normalized size = 1.02 \[ \frac {\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{b} \]
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 \[ x \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} + 2 \, b \int \frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} x \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 44, normalized size = 0.94 \[ \frac {\left ({\mathrm {asin}\left (a+b\,x\right )}^2-2\right )\,\left (a+b\,x\right )}{b}+\frac {2\,\mathrm {asin}\left (a+b\,x\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 63, normalized size = 1.34 \[ \begin {cases} \frac {a \operatorname {asin}^{2}{\left (a + b x \right )}}{b} + x \operatorname {asin}^{2}{\left (a + b x \right )} - 2 x + \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asin}^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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