Optimal. Leaf size=118 \[ \frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6470, 30, 265,
272, 44, 65, 214} \begin {gather*} \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )+\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 44
Rule 65
Rule 214
Rule 265
Rule 272
Rule 6470
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x^3} \, dx &=-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\int \frac {1}{x^5} \, dx}{a}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^5 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^5 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}-\frac {1}{8} \left (a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^4\right )\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^4}\right )}{4 a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 105, normalized size = 0.89 \begin {gather*} -\frac {\frac {1}{x^4}+\frac {\sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )}{x^4}-\frac {a^2 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right ) \text {ArcTan}\left (\sqrt {-1+a^2 x^4}\right )}{\sqrt {-1+a^2 x^4}}}{4 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.08, size = 129, normalized size = 1.09
method | result | size |
default | \(-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (-\ln \left (\frac {2 \,\mathrm {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}+2}{a^{2} x^{2}}\right ) x^{4} a +\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, \mathrm {csgn}\left (\frac {1}{a}\right )\right ) \mathrm {csgn}\left (\frac {1}{a}\right )}{4 x^{2} \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}-\frac {1}{4 a \,x^{4}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 146, normalized size = 1.24 \begin {gather*} \frac {a^{2} x^{4} \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1\right ) - a^{2} x^{4} \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1\right ) - 2 \, a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 2}{8 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{x^{5}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x^{3}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.02, size = 71, normalized size = 0.60 \begin {gather*} \frac {a\,\ln \left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right )}{4}-\frac {1}{4\,a\,x^4}-\frac {\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}}{4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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