Optimal. Leaf size=110 \[ \frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(1-a x)^{5/4} (a x+1)^{3/4}}{2 x^2}+\frac {a \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x} \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6126, 96, 94, 93, 298, 203, 206} \[ \frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(1-a x)^{5/4} (a x+1)^{3/4}}{2 x^2}+\frac {a \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 298
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{-\frac {1}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {\sqrt [4]{1-a x}}{x^3 \sqrt [4]{1+a x}} \, dx\\ &=-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}-\frac {1}{4} a \int \frac {\sqrt [4]{1-a x}}{x^2 \sqrt [4]{1+a x}} \, dx\\ &=\frac {a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac {1}{8} a^2 \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac {a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac {1}{2} a^2 \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 {a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}-\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 69, normalized size = 0.63 \[ \frac {\sqrt [4]{1-a x} \left (-2 a^2 x^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{a x+1}\right )+3 a^2 x^2+a x-2\right )}{4 x^2 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 148, normalized size = 1.35 \[ \frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, a x - 2\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}} \]
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 \[ \int \frac {1}{x^{3} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{3}}\, dx \]
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 \[ \int \frac {1}{x^{3} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \]
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 \[ \int \frac {1}{x^{3} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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