Optimal. Leaf size=82 \[ -\frac {6 \sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}+\frac {3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}-\frac {6 (a+b x) \sin ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4803, 4619, 4677, 261} \[ -\frac {6 \sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}+\frac {3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}-\frac {6 (a+b x) \sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \sin ^{-1}(a+b x)^3 \, dx &=\frac {\operatorname {Subst}\left (\int \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}-\frac {3 \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}-\frac {6 \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac {6 (a+b x) \sin ^{-1}(a+b x)}{b}+\frac {3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}+\frac {6 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \sin ^{-1}(a+b x)}{b}+\frac {3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^3}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.90 \[ \frac {-6 \sqrt {1-(a+b x)^2}+(a+b x) \sin ^{-1}(a+b x)^3+3 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2-6 (a+b x) \sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 66, normalized size = 0.80 \[ \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3} - 6 \, {\left (b x + a\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\arcsin \left (b x + a\right )^{2} - 2\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 78, normalized size = 0.95 \[ \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{b} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b} - \frac {6 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )}{b} - \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 71, normalized size = 0.87 \[ \frac {\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arcsin \left (b x +a \right )}{b} \]
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 \[ x \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{3} + 3 \, b \int \frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} x \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 59, normalized size = 0.72 \[ \frac {\left (3\,{\mathrm {asin}\left (a+b\,x\right )}^2-6\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b}-\frac {\left (6\,\mathrm {asin}\left (a+b\,x\right )-{\mathrm {asin}\left (a+b\,x\right )}^3\right )\,\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 109, normalized size = 1.33 \[ \begin {cases} \frac {a \operatorname {asin}^{3}{\left (a + b x \right )}}{b} - \frac {6 a \operatorname {asin}{\left (a + b x \right )}}{b} + x \operatorname {asin}^{3}{\left (a + b x \right )} - 6 x \operatorname {asin}{\left (a + b x \right )} + \frac {3 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{b} - \frac {6 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asin}^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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