Optimal. Leaf size=110 \[ -\frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {9}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(1-a x)^{7/4} \sqrt [4]{a x+1}}{2 x^2}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x} \]
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Rubi [A] time = 0.04, 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, 212, 206, 203} \[ -\frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {9}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(1-a x)^{7/4} \sqrt [4]{a x+1}}{2 x^2}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{-\frac {3}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {(1-a x)^{3/4}}{x^3 (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac {1}{4} (3 a) \int \frac {(1-a x)^{3/4}}{x^2 (1+a x)^{3/4}} \, dx\\ &=\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}+\frac {1}{8} \left (9 a^2\right ) \int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}+\frac {1}{2} \left (9 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac {1}{4} \left (9 a^2\right ) \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} \left (9 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 x^2}-\frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {9}{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 = 70, normalized size = 0.64 \[ \frac {(1-a x)^{3/4} \left (-6 a^2 x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{a x+1}\right )+5 a^2 x^2+3 a x-2\right )}{4 x^2 (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 145, normalized size = 1.32 \[ -\frac {18 \, a^{2} x^{2} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 9 \, a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 9 \, a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, {\left (5 \, a^{2} x^{2} - 7 \, a x + 2\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}} \]
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 \[ \int \frac {1}{x^{3} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{3}}\, dx \]
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 \[ \int \frac {1}{x^{3} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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