Optimal. Leaf size=147 \[ -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{a x}+x e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2}{a x} \]
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Rubi [A] time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6330, 30, 259, 325, 307, 221, 1199, 424} \[ -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{a x}+x e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2}{a x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 221
Rule 259
Rule 307
Rule 325
Rule 424
Rule 1199
Rule 6330
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx &=e^{\text {sech}^{-1}\left (a x^2\right )} x+\frac {2 \int \frac {1}{x^2} \, dx}{a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^2 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=-\frac {2}{a x}+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^2 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=-\frac {2}{a x}+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}}{a x}-\left (2 a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {2}{a x}+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}}{a x}+\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx-\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1+a x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {2}{a x}+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}}{a x}+\frac {2 \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 )}{\sqrt {a}}-\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1+a x^2}}{\sqrt {1-a x^2}} \, dx\\ &=-\frac {2}{a x}+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}}{a x}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+\frac {2 \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 )}{\sqrt {a}}\\ \end {align*}
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Mathematica [C] time = 0.30, size = 135, normalized size = 0.92 \[ -\frac {2 i \sqrt {\frac {1-a x^2}{a x^2+1}} \sqrt {1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )\right )}{\sqrt {-a} \left (a x^2-1\right )}+\sqrt {\frac {1-a x^2}{a x^2+1}} \left (-\frac {1}{a x}-x\right )-\frac {1}{a x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{a x^{2}}, x\right ) \]
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 \[ \int \sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 132, normalized size = 0.90 \[ -\frac {1}{a x}-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{2} x^{4}+2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {a}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \EllipticE \left (x \sqrt {a}, i\right ) \sqrt {a}-1\right )}{a^{2} x^{4}-1} \]
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 \[ \frac {\int \frac {\sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}}{x^{2}}\,{d x}}{a} - \frac {1}{a x} \]
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 \[ \int \sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2} \,d x \]
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 \[ \frac {\int \frac {1}{x^{2}}\, dx + \int a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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