Optimal. Leaf size=54 \[ \frac {1}{6 x^2}-\frac {2 \sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac {2 \log (x)}{3} \]
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Rubi [A] time = 0.09, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4702, 4682, 29, 30} \[ \frac {1}{6 x^2}-\frac {2 \sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac {2 \log (x)}{3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 30
Rule 4682
Rule 4702
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(x)}{x^4 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac {1}{3} \int \frac {1}{x^3} \, dx+\frac {2}{3} \int \frac {\cos ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx\\ &=\frac {1}{6 x^2}-\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac {2 \sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {2}{3} \int \frac {1}{x} \, dx\\ &=\frac {1}{6 x^2}-\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac {2 \sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {2 \log (x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 0.70 \[ \frac {-4 x^3 \log (x)-2 \sqrt {1-x^2} \left (2 x^2+1\right ) \cos ^{-1}(x)+x}{6 x^3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{-1}(x)}{x^4 \sqrt {1-x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.08, size = 36, normalized size = 0.67 \[ -\frac {4 \, x^{3} \log \relax (x) + 2 \, {\left (2 \, x^{2} + 1\right )} \sqrt {-x^{2} + 1} \arccos \relax (x) - x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 95, normalized size = 1.76 \[ \frac {1}{24} \, {\left (\frac {x^{3} {\left (\frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \arccos \relax (x) + \frac {2 \, x^{2} + 1}{6 \, x^{2}} - \frac {1}{3} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 43, normalized size = 0.80
method | result | size |
default | \(\frac {1}{6 x^{2}}-\frac {2 \ln \relax (x )}{3}-\frac {\arccos \relax (x ) \sqrt {-x^{2}+1}}{3 x^{3}}-\frac {2 \arccos \relax (x ) \sqrt {-x^{2}+1}}{3 x}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 42, normalized size = 0.78 \[ -\frac {1}{3} \, {\left (\frac {2 \, \sqrt {-x^{2} + 1}}{x} + \frac {\sqrt {-x^{2} + 1}}{x^{3}}\right )} \arccos \relax (x) + \frac {1}{6 \, x^{2}} - \frac {2}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acos}\relax (x)}{x^4\,\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 139.93, size = 60, normalized size = 1.11 \[ \left (\begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {\left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {acos}{\relax (x )} + \begin {cases} \text {NaN} & \text {for}\: x < -1 \\- \frac {2 \log {\relax (x )}}{3} - \frac {1}{6} + \frac {2 i \pi }{3} + \frac {1}{6 x^{2}} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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