Optimal. Leaf size=36 \[ \frac {(a+b x) \cos ^{-1}(a+b x)}{b}-\frac {\sqrt {1-(a+b x)^2}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4804, 4620, 261} \[ \frac {(a+b x) \cos ^{-1}(a+b x)}{b}-\frac {\sqrt {1-(a+b x)^2}}{b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4620
Rule 4804
Rubi steps
\begin {align*} \int \cos ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \cos ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cos ^{-1}(a+b x)}{b}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \cos ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 1.31 \[ x \cos ^{-1}(a+b x)-\frac {\sqrt {-a^2-2 a b x-b^2 x^2+1}+a \sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 41, normalized size = 1.14 \[ \frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.98, size = 32, normalized size = 0.89 \[ \frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.92 \[ \frac {\left (b x +a \right ) \arccos \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 32, normalized size = 0.89 \[ \frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 88, normalized size = 2.44 \[ x\,\mathrm {acos}\left (a+b\,x\right )-\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}-\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 46, normalized size = 1.28 \[ \begin {cases} \frac {a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}{\left (a + b x \right )} - \frac {\sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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