Optimal. Leaf size=115 \[ \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{3 a x^3}+\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {2}{3} \sqrt {a} \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6335, 30, 259, 325, 221} \[ \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{3 a x^3}+\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {2}{3} \sqrt {a} \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 30
Rule 221
Rule 259
Rule 325
Rule 6335
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x^2} \, dx &=-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {2 \int \frac {1}{x^4} \, dx}{a}-\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^4 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^4 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {1}{3} \left (2 a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {2}{3} \sqrt {a} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.20, size = 123, normalized size = 1.07 \[ \frac {2 i \sqrt {-a} \sqrt {\frac {1-a x^2}{a x^2+1}} \sqrt {1-a^2 x^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{3 a x^2-3}-\frac {1}{3 a x^3}-\frac {\sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right )}{3 a x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{a x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 104, normalized size = 0.90 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \EllipticF \left (x \sqrt {a}, i\right ) x^{3} a^{\frac {3}{2}}-a^{2} x^{4}+1\right )}{3 x \left (a^{2} x^{4}-1\right )}-\frac {1}{3 x^{3} a} \]
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 \[ \frac {\int \frac {\sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}}{x^{4}}\,{d x}}{a} - \frac {1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}}{x^2} \,d x \]
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 \[ \frac {\int \frac {1}{x^{4}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x^{2}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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