Integrand size = 10, antiderivative size = 56 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{3 a^3}-\frac {\left (1-\frac {a^2}{x^2}\right )^{3/2}}{9 a^3}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4917, 5328, 272, 45} \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {\left (1-\frac {a^2}{x^2}\right )^{3/2}}{9 a^3}+\frac {\sqrt {1-\frac {a^2}{x^2}}}{3 a^3}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3} \]
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Rule 45
Rule 272
Rule 4917
Rule 5328
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x^4} \, dx \\ & = -\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3}+\frac {1}{3} a \int \frac {1}{\sqrt {1-\frac {a^2}{x^2}} x^5} \, dx \\ & = -\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {1}{6} a \text {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {\sqrt {1-\frac {a^2}{x^2}}}{3 a^3}-\frac {\left (1-\frac {a^2}{x^2}\right )^{3/2}}{9 a^3}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{3 x^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}} x \left (a^2+2 x^2\right )-3 a^3 \arccos \left (\frac {a}{x}\right )}{9 a^3 x^3} \]
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Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(-\frac {\frac {a^{3} \arccos \left (\frac {a}{x}\right )}{3 x^{3}}-\frac {a^{2} \sqrt {1-\frac {a^{2}}{x^{2}}}}{9 x^{2}}-\frac {2 \sqrt {1-\frac {a^{2}}{x^{2}}}}{9}}{a^{3}}\) | \(55\) |
default | \(-\frac {\frac {a^{3} \arccos \left (\frac {a}{x}\right )}{3 x^{3}}-\frac {a^{2} \sqrt {1-\frac {a^{2}}{x^{2}}}}{9 x^{2}}-\frac {2 \sqrt {1-\frac {a^{2}}{x^{2}}}}{9}}{a^{3}}\) | \(55\) |
parts | \(-\frac {\arccos \left (\frac {a}{x}\right )}{3 x^{3}}-\frac {\left (a^{2}-x^{2}\right ) \left (a^{2}+2 x^{2}\right )}{9 a^{3} \sqrt {-\frac {a^{2}-x^{2}}{x^{2}}}\, x^{4}}\) | \(55\) |
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {3 \, a^{3} \arccos \left (\frac {a}{x}\right ) - {\left (a^{2} x + 2 \, x^{3}\right )} \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}}}{9 \, a^{3} x^{3}} \]
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Time = 1.71 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.79 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=\frac {a \left (\begin {cases} \frac {\sqrt {-1 + \frac {x^{2}}{a^{2}}}}{3 a x^{3}} + \frac {2 \sqrt {-1 + \frac {x^{2}}{a^{2}}}}{3 a^{3} x} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\\frac {i \sqrt {1 - \frac {x^{2}}{a^{2}}}}{3 a x^{3}} + \frac {2 i \sqrt {1 - \frac {x^{2}}{a^{2}}}}{3 a^{3} x} & \text {otherwise} \end {cases}\right )}{3} - \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{3 x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {1}{9} \, a {\left (\frac {{\left (-\frac {a^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}}}{a^{4}} - \frac {3 \, \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a^{4}}\right )} - \frac {\arccos \left (\frac {a}{x}\right )}{3 \, x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {\frac {3 \, a \arccos \left (\frac {a}{x}\right )}{x^{3}} - \frac {2 \, \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a^{2}} - \frac {\sqrt {-\frac {a^{2}}{x^{2}} + 1}}{x^{2}}}{9 \, a} \]
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Timed out. \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^4} \, dx=\int \frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x^4} \,d x \]
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