Optimal. Leaf size=110 \[ -\frac {3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 x^2}+\frac {9}{4} a^2 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {9}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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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 {9}{4} a^2 \text {ArcTan}\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 {\sqrt [4]{1-a x} (a x+1)^{7/4}}{2 x^2}-\frac {3 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 {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 {\sqrt [4]{1-a x} (1+a x)^{7/4}}{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 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 x^2}+\frac {1}{8} \left (9 a^2\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 x^2}+\frac {1}{2} \left (9 a^2\right ) \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 {3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 x^2}-\frac {1}{4} \left (9 a^2\right ) \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} \left (9 a^2\right ) \text {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 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 70, normalized size = 0.64 \begin {gather*} -\frac {\sqrt [4]{1-a x} \left (2+7 a x+5 a^2 x^2+18 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 \[\int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}{x^{3}}\, dx\]
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 \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 = 0.38, size = 149, normalized size = 1.35 \begin {gather*} \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 \, \sqrt {-a^{2} x^{2} + 1} {\left (5 \, 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 {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{3}}\, 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 {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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