Integrand size = 6, antiderivative size = 36 \[ \int \arccos (a+b x) \, dx=-\frac {\sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \arccos (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4716, 267} \[ \int \arccos (a+b x) \, dx=\frac {(a+b x) \arccos (a+b x)}{b}-\frac {\sqrt {1-(a+b x)^2}}{b} \]
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Rule 267
Rule 4716
Rule 4888
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \arccos (x) \, dx,x,a+b x)}{b} \\ & = \frac {(a+b x) \arccos (a+b x)}{b}+\frac {\text {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) \arccos (a+b x)}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(36)=72\).
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 4.28 \[ \int \arccos (a+b x) \, dx=x \arccos (a+b x)-\frac {2 b \sqrt {1-a^2-2 a b x-b^2 x^2}+2 a b \arctan \left (\frac {\sqrt {-b^2} x-\sqrt {1-a^2-2 a b x-b^2 x^2}}{a}\right )+a \sqrt {-b^2} \log \left (-1+2 a b x+2 b^2 x^2+2 \sqrt {-b^2} x \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{2 b^2} \]
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Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \arccos \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{b}\) | \(33\) |
default | \(\frac {\left (b x +a \right ) \arccos \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{b}\) | \(33\) |
parts | \(x \arccos \left (b x +a \right )+b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )\) | \(87\) |
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Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
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Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \arccos (a+b x) \, dx=\begin {cases} \frac {a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}{\left (a + b x \right )} - \frac {\sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
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Time = 0.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.44 \[ \int \arccos (a+b x) \, dx=x\,\mathrm {acos}\left (a+b\,x\right )-\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}-\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
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