Optimal. Leaf size=60 \[ \frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a^3}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5264, 4628, 266, 51, 63, 208} \[ \frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a^3}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 4628
Rule 5264
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx &=\int \frac {\cos ^{-1}\left (\frac {x}{a}\right )}{x^4} \, dx\\ &=-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx}{3 a}\\ &=-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{6 a}\\ &=\frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{12 a^3}\\ &=\frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a}\\ &=\frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{3 x^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 1.15 \[ \frac {-2 a^3 \sec ^{-1}\left (\frac {a}{x}\right )+a^2 x \sqrt {1-\frac {x^2}{a^2}}+x^3 \log \left (\sqrt {1-\frac {x^2}{a^2}}+1\right )-x^3 \log (x)}{6 a^3 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 142, normalized size = 2.37 \[ -\frac {4 \, a^{3} x^{3} \arctan \left (-\frac {x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{a^{2} - x^{2}}\right ) - x^{3} \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} + a\right ) + x^{3} \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - a\right ) - 2 \, a x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - 4 \, {\left (a^{3} x^{3} - a^{3}\right )} \operatorname {arcsec}\left (\frac {a}{x}\right )}{12 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 80, normalized size = 1.33 \[ \frac {a {\left (\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a^{3}} - \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a^{3}} + \frac {2 \, \sqrt {a^{2} - x^{2}}}{a^{2} x^{2}}\right )}}{12 \, {\left | a \right |}} - \frac {\arccos \left (\frac {x}{a}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 98, normalized size = 1.63 \[ -\frac {\mathrm {arcsec}\left (\frac {a}{x}\right )}{3 x^{3}}+\frac {-1+\frac {a^{2}}{x^{2}}}{6 a^{3} \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}}+\frac {\sqrt {-1+\frac {a^{2}}{x^{2}}}\, x \ln \left (\frac {a}{x}+\sqrt {-1+\frac {a^{2}}{x^{2}}}\right )}{6 a^{4} \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 64, normalized size = 1.07 \[ \frac {\frac {\log \left (\frac {2 \, \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a^{2}} + \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{x^{2}}}{6 \, a} - \frac {\operatorname {arcsec}\left (\frac {a}{x}\right )}{3 \, x^{3}} \]
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 \[ \int \frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.52, size = 100, normalized size = 1.67 \[ - \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{3 x^{3}} - \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{2 a x} - \frac {\operatorname {acosh}{\left (\frac {a}{x} \right )}}{2 a^{2}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\\frac {i a}{2 x^{3} \sqrt {- \frac {a^{2}}{x^{2}} + 1}} - \frac {i}{2 a x \sqrt {- \frac {a^{2}}{x^{2}} + 1}} + \frac {i \operatorname {asin}{\left (\frac {a}{x} \right )}}{2 a^{2}} & \text {otherwise} \end {cases}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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