Optimal. Leaf size=80 \[ \frac{\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{1-(a+b x)^2}}{4 b^2}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)-\frac{x \sqrt{1-(a+b x)^2}}{4 b} \]
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Rubi [A] time = 0.0731415, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4806, 4744, 743, 641, 216} \[ \frac{\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{1-(a+b x)^2}}{4 b^2}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)-\frac{x \sqrt{1-(a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 4806
Rule 4744
Rule 743
Rule 641
Rule 216
Rubi steps
\begin{align*} \int x \cos ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right ) \cos ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{2} x^2 \cos ^{-1}(a+b x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac{x \sqrt{1-(a+b x)^2}}{4 b}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{1+2 a^2}{b^2}+\frac{3 a x}{b^2}}{\sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=\frac{3 a \sqrt{1-(a+b x)^2}}{4 b^2}-\frac{x \sqrt{1-(a+b x)^2}}{4 b}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)+\frac{\left (1+2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{4 b^2}\\ &=\frac{3 a \sqrt{1-(a+b x)^2}}{4 b^2}-\frac{x \sqrt{1-(a+b x)^2}}{4 b}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)+\frac{\left (1+2 a^2\right ) \sin ^{-1}(a+b x)}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0494883, size = 69, normalized size = 0.86 \[ \frac{(3 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+\left (2 a^2+1\right ) \sin ^{-1}(a+b x)+2 b^2 x^2 \cos ^{-1}(a+b x)}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 78, normalized size = 1. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{\arccos \left ( bx+a \right ) \left ( bx+a \right ) ^{2}}{2}}-\arccos \left ( bx+a \right ) a \left ( bx+a \right ) -{\frac{bx+a}{4}\sqrt{1- \left ( bx+a \right ) ^{2}}}+{\frac{\arcsin \left ( bx+a \right ) }{4}}+a\sqrt{1- \left ( bx+a \right ) ^{2}} \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 = 2.55125, size = 135, normalized size = 1.69 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x - 3 \, a\right )}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.361235, size = 104, normalized size = 1.3 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{acos}{\left (a + b x \right )}}{2 b^{2}} + \frac{3 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b^{2}} + \frac{x^{2} \operatorname{acos}{\left (a + b x \right )}}{2} - \frac{x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b} - \frac{\operatorname{acos}{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{acos}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32168, size = 119, normalized size = 1.49 \begin{align*} \frac{{\left (b x + a\right )}^{2} \arccos \left (b x + a\right )}{2 \, b^{2}} - \frac{{\left (b x + a\right )} a \arccos \left (b x + a\right )}{b^{2}} - \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )}}{4 \, b^{2}} + \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac{\arccos \left (b x + a\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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