Optimal. Leaf size=29 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 63, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-b+a x}} \, dx,x,x^3\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {-b+a x^3}\right )}{3 a}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 64, normalized size = 2.21 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{3 \, b}, \frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 21, normalized size = 0.72 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.90
method | result | size |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\) | \(26\) |
elliptic | \(-\frac {2 \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 21, normalized size = 0.72 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 37, normalized size = 1.28 \begin {gather*} \frac {\ln \left (\frac {a\,x^3-2\,b+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{3\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 60, normalized size = 2.07 \begin {gather*} \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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