Optimal. Leaf size=128 \[ \frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{4 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{2}{3} d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{27} b^2 c^2 d x^3-\frac{14}{9} b^2 d x \]
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Rubi [A] time = 0.137281, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4649, 4619, 4677, 8} \[ \frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{4 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{2}{3} d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{27} b^2 c^2 d x^3-\frac{14}{9} b^2 d x \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4619
Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} (2 d) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{3} (2 b c d) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{2}{3} d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{9} \left (2 b^2 d\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac{1}{3} (4 b c d) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{2}{9} b^2 d x+\frac{2}{27} b^2 c^2 d x^3+\frac{4 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{2}{3} d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{3} \left (4 b^2 d\right ) \int 1 \, dx\\ &=-\frac{14}{9} b^2 d x+\frac{2}{27} b^2 c^2 d x^3+\frac{4 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{2}{3} d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.203339, size = 137, normalized size = 1.07 \[ -\frac{d \left (9 a^2 c x \left (c^2 x^2-3\right )+6 a b \sqrt{1-c^2 x^2} \left (c^2 x^2-7\right )+6 b \sin ^{-1}(c x) \left (3 a c x \left (c^2 x^2-3\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2-7\right )\right )-2 b^2 c x \left (c^2 x^2-21\right )+9 b^2 c x \left (c^2 x^2-3\right ) \sin ^{-1}(c x)^2\right )}{27 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 173, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ( -d{a}^{2} \left ({\frac{{c}^{3}{x}^{3}}{3}}-cx \right ) -d{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-3 \right ) cx}{3}}+{\frac{4\,cx}{3}}-{\frac{4\,\arcsin \left ( cx \right ) }{3}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 2\,{c}^{2}{x}^{2}-6 \right ) cx}{27}} \right ) -2\,dab \left ( 1/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) -cx\arcsin \left ( cx \right ) +1/9\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{7\,\sqrt{-{c}^{2}{x}^{2}+1}}{9}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6771, size = 315, normalized size = 2.46 \begin{align*} -\frac{1}{3} \, b^{2} c^{2} d x^{3} \arcsin \left (c x\right )^{2} - \frac{1}{3} \, a^{2} c^{2} d x^{3} - \frac{2}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d - \frac{2}{27} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \arcsin \left (c x\right )^{2} - 2 \, b^{2} d{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81685, size = 335, normalized size = 2.62 \begin{align*} -\frac{{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} d x^{3} - 3 \,{\left (9 \, a^{2} - 14 \, b^{2}\right )} c d x + 9 \,{\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (a b c^{3} d x^{3} - 3 \, a b c d x\right )} \arcsin \left (c x\right ) + 6 \,{\left (a b c^{2} d x^{2} - 7 \, a b d +{\left (b^{2} c^{2} d x^{2} - 7 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{27 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.74528, size = 224, normalized size = 1.75 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} d x^{3}}{3} + a^{2} d x - \frac{2 a b c^{2} d x^{3} \operatorname{asin}{\left (c x \right )}}{3} - \frac{2 a b c d x^{2} \sqrt{- c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname{asin}{\left (c x \right )} + \frac{14 a b d \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{b^{2} c^{2} d x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} + \frac{2 b^{2} c^{2} d x^{3}}{27} - \frac{2 b^{2} c d x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9} + b^{2} d x \operatorname{asin}^{2}{\left (c x \right )} - \frac{14 b^{2} d x}{9} + \frac{14 b^{2} d \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c} & \text{for}\: c \neq 0 \\a^{2} d x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45402, size = 265, normalized size = 2.07 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} d x^{3} - \frac{1}{3} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2} - \frac{2}{3} \,{\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right ) + \frac{2}{3} \, b^{2} d x \arcsin \left (c x\right )^{2} + \frac{2}{27} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d x + \frac{4}{3} \, a b d x \arcsin \left (c x\right ) + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d \arcsin \left (c x\right )}{9 \, c} + a^{2} d x - \frac{40}{27} \, b^{2} d x + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d}{9 \, c} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{3 \, c} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} a b d}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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