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 = 0.118784, antiderivative size = 76, normalized size of antiderivative = 1., 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 4627
Rule 4707
Rule 4641
Rule 30
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*}
Mathematica [A] time = 0.0734169, 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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Maple [A] time = 0.023, size = 120, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{{a}^{2}{c}^{2}{x}^{2}}{2}}+{b}^{2} \left ({\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{2}}+{\frac{\arcsin \left ( cx \right ) }{2} \left ( cx\sqrt{-{c}^{2}{x}^{2}+1}+\arcsin \left ( cx \right ) \right ) }-{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{4}}-{\frac{{c}^{2}{x}^{2}}{4}} \right ) +2\,ab \left ( 1/2\,{c}^{2}{x}^{2}\arcsin \left ( cx \right ) +1/4\,cx\sqrt{-{c}^{2}{x}^{2}+1}-1/4\,\arcsin \left ( cx \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48231, size = 221, normalized size = 2.91 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.17421, size = 126, normalized size = 1.66 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29979, size = 209, normalized size = 2.75 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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