Optimal. Leaf size=72 \[ -\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}-a \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6126, 94, 93, 298, 203, 206} \[ -\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}-a \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 203
Rule 206
Rule 298
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{-\frac {1}{2} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {\sqrt [4]{1-a x}}{x^2 \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-\frac {1}{2} a \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-(2 a) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}+a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-a \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-a \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 0.76 \[ \frac {\sqrt [4]{1-a x} \left (2 a x \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{a x+1}\right )-a x-1\right )}{x \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.51, size = 130, normalized size = 1.81 \[ -\frac {2 \, a x \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x} \]
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 \[ \int \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{2}}\, dx \]
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 \[ \int \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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