Optimal. Leaf size=82 \[ \frac{6 \sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}-\frac{6 (a+b x) \cos ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.0820063, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4804, 4620, 4678, 261} \[ \frac{6 \sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}-\frac{6 (a+b x) \cos ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4804
Rule 4620
Rule 4678
Rule 261
Rubi steps
\begin{align*} \int \cos ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \cos ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}+\frac{3 \operatorname{Subst}\left (\int \frac{x \cos ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac{6 \operatorname{Subst}\left (\int \cos ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac{6 (a+b x) \cos ^{-1}(a+b x)}{b}-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \cos ^{-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) \cos ^{-1}(a+b x)}{b}-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}\\ \end{align*}
Mathematica [A] time = 0.0356647, size = 74, normalized size = 0.9 \[ \frac{6 \sqrt{1-(a+b x)^2}+(a+b x) \cos ^{-1}(a+b x)^3-3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2-6 (a+b x) \cos ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 71, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( \arccos \left ( bx+a \right ) \right ) ^{3} \left ( bx+a \right ) -3\, \left ( \arccos \left ( bx+a \right ) \right ) ^{2}\sqrt{1- \left ( bx+a \right ) ^{2}}+6\,\sqrt{1- \left ( bx+a \right ) ^{2}}-6\, \left ( bx+a \right ) \arccos \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55929, size = 170, normalized size = 2.07 \begin{align*} \frac{{\left (b x + a\right )} \arccos \left (b x + a\right )^{3} - 6 \,{\left (b x + a\right )} \arccos \left (b x + a\right ) - 3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (\arccos \left (b x + a\right )^{2} - 2\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.692653, size = 109, normalized size = 1.33 \begin{align*} \begin{cases} \frac{a \operatorname{acos}^{3}{\left (a + b x \right )}}{b} - \frac{6 a \operatorname{acos}{\left (a + b x \right )}}{b} + x \operatorname{acos}^{3}{\left (a + b x \right )} - 6 x \operatorname{acos}{\left (a + b x \right )} - \frac{3 \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{acos}^{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{acos}^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42999, size = 105, normalized size = 1.28 \begin{align*} \frac{{\left (b x + a\right )} \arccos \left (b x + a\right )^{3}}{b} - \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} \arccos \left (b x + a\right )^{2}}{b} - \frac{6 \,{\left (b x + a\right )} \arccos \left (b x + a\right )}{b} + \frac{6 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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