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 = 0.0909131, antiderivative size = 54, normalized size of antiderivative = 1., 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 4702
Rule 4682
Rule 29
Rule 30
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*}
Mathematica [A] time = 0.0430401, size = 38, normalized size = 0.7 \[ \frac{-4 x^3 \log (x)-2 \sqrt{1-x^2} \left (2 x^2+1\right ) \cos ^{-1}(x)+x}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 43, normalized size = 0.8 \begin{align*}{\frac{1}{6\,{x}^{2}}}-{\frac{2\,\ln \left ( x \right ) }{3}}-{\frac{\arccos \left ( x \right ) }{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{2\,\arccos \left ( x \right ) }{3\,x}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4098, size = 57, normalized size = 1.06 \begin{align*} -\frac{1}{3} \,{\left (\frac{2 \, \sqrt{-x^{2} + 1}}{x} + \frac{\sqrt{-x^{2} + 1}}{x^{3}}\right )} \arccos \left (x\right ) + \frac{1}{6 \, x^{2}} - \frac{2}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69828, size = 95, normalized size = 1.76 \begin{align*} -\frac{4 \, x^{3} \log \left (x\right ) + 2 \,{\left (2 \, x^{2} + 1\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 114.909, size = 60, normalized size = 1.11 \begin{align*} \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}{\left (x \right )} + \begin{cases} \text{NaN} & \text{for}\: x < -1 \\- \frac{2 \log{\left (x \right )}}{3} - \frac{1}{6} + \frac{2 i \pi }{3} + \frac{1}{6 x^{2}} & \text{for}\: x < 1 \\\text{NaN} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10374, size = 128, normalized size = 2.37 \begin{align*} \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 \left (x\right ) + \frac{2 \, x^{2} + 1}{6 \, x^{2}} - \frac{1}{3} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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