Optimal. Leaf size=76 \[ \frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{4} b^2 x^2 \]
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Rubi [A] time = 0.12, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4627, 4707, 4641, 30} \[ \frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{4} b^2 x^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 4627
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^2-(b c) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{2} b^2 \int x \, dx-\frac {b \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}\\ &=-\frac {1}{4} b^2 x^2+\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.08, size = 73, normalized size = 0.96 \[ -\frac {-2 c^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2-2 b c x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\left (a+b \sin ^{-1}(c x)\right )^2+b^2 c^2 x^2}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 99, normalized size = 1.30 \[ \frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} x^{2} + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 155, normalized size = 2.04 \[ \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right )}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b x}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2}}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2}}{4 \, c^{2}} + \frac {a b \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2}}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 120, normalized size = 1.58 \[ \frac {\frac {a^{2} c^{2} x^{2}}{2}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{2}+\frac {\arcsin \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2}-\frac {\arcsin \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}\right )+2 a b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}} \]
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 \[ \frac {1}{2} \, a^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, c \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} x^{2} - 1}\,{d x}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 126, normalized size = 1.66 \[ \begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {asin}{\left (c x \right )} + \frac {a b x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} x^{2}}{4} + \frac {b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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