Integrand size = 8, antiderivative size = 58 \[ \int \frac {\arccos (a x)}{x^5} \, dx=\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{6 x}-\frac {\arccos (a x)}{4 x^4} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 277, 270} \[ \int \frac {\arccos (a x)}{x^5} \, dx=\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{6 x}-\frac {\arccos (a x)}{4 x^4} \]
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Rule 270
Rule 277
Rule 4724
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)}{4 x^4}-\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2}}{12 x^3}-\frac {\arccos (a x)}{4 x^4}-\frac {1}{6} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{6 x}-\frac {\arccos (a x)}{4 x^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int \frac {\arccos (a x)}{x^5} \, dx=\frac {a x \sqrt {1-a^2 x^2} \left (1+2 a^2 x^2\right )-3 \arccos (a x)}{12 x^4} \]
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Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90
method | result | size |
parts | \(-\frac {\arccos \left (a x \right )}{4 x^{4}}-\frac {a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )}{4}\) | \(52\) |
derivativedivides | \(a^{4} \left (-\frac {\arccos \left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {-a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a x}\right )\) | \(58\) |
default | \(a^{4} \left (-\frac {\arccos \left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {-a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a x}\right )\) | \(58\) |
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none
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.64 \[ \int \frac {\arccos (a x)}{x^5} \, dx=\frac {{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} - 3 \, \arccos \left (a x\right )}{12 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 1.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.76 \[ \int \frac {\arccos (a x)}{x^5} \, dx=- \frac {a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{4} - \frac {\operatorname {acos}{\left (a x \right )}}{4 x^{4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {\arccos (a x)}{x^5} \, dx=\frac {1}{12} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac {\arccos \left (a x\right )}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (48) = 96\).
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.24 \[ \int \frac {\arccos (a x)}{x^5} \, dx=-\frac {1}{96} \, {\left (\frac {{\left (a^{4} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2} {\left | a \right |}}\right )} a - \frac {\arccos \left (a x\right )}{4 \, x^{4}} \]
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Timed out. \[ \int \frac {\arccos (a x)}{x^5} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x^5} \,d x \]
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