Optimal. Leaf size=110 \[ \frac {a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 x^2}+\frac {1}{4} a^2 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.03, 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 = {6261, 98, 96,
95, 304, 209, 212} \begin {gather*} \frac {1}{4} a^2 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 209
Rule 212
Rule 304
Rule 6261
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 \text {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 \text {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 \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 69, normalized size = 0.63 \begin {gather*} \frac {\sqrt [4]{1-a x} \left (-2+a x+3 a^2 x^2-2 a^2 x^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{1+a x}\right )\right )}{4 x^2 \sqrt [4]{1+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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 = 0.36, size = 148, normalized size = 1.35 \begin {gather*} \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}} \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*} \int \frac {1}{x^{3} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^3\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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