Integrand size = 12, antiderivative size = 39 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {a+b \arccos (c x)}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4724, 270} \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {a+b \arccos (c x)}{2 x^2} \]
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Rule 270
Rule 4724
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arccos (c x)}{2 x^2}-\frac {1}{2} (b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {a+b \arccos (c x)}{2 x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=-\frac {a}{2 x^2}+\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {b \arccos (c x)}{2 x^2} \]
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Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18
method | result | size |
parts | \(-\frac {a}{2 x^{2}}+b \,c^{2} \left (-\frac {\arccos \left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\) | \(46\) |
derivativedivides | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\arccos \left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\) | \(50\) |
default | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\arccos \left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=\frac {\sqrt {-c^{2} x^{2} + 1} b c x + a x^{2} - b \arccos \left (c x\right ) - a}{2 \, x^{2}} \]
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Time = 0.82 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=- \frac {a}{2 x^{2}} - \frac {b c \left (\begin {cases} - \frac {i \sqrt {c^{2} x^{2} - 1}}{x} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- c^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \operatorname {acos}{\left (c x \right )}}{2 x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=\frac {1}{2} \, b {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} - \frac {\arccos \left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (33) = 66\).
Time = 0.29 (sec) , antiderivative size = 492, normalized size of antiderivative = 12.62 \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=-\frac {b c^{2} \arccos \left (c x\right )}{2 \, {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac {a c^{2}}{2 \, {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} b c^{2} \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac {\sqrt {-c^{2} x^{2} + 1} b c^{2}}{{\left (c x + 1\right )} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} a c^{2}}{{\left (c x + 1\right )}^{2} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b c^{2} \arccos \left (c x\right )}{2 \, {\left (c x + 1\right )}^{4} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b c^{2}}{{\left (c x + 1\right )}^{3} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a c^{2}}{2 \, {\left (c x + 1\right )}^{4} {\left (\frac {2 \, {\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{x^3} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3} \,d x \]
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