Optimal. Leaf size=139 \[ -\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac {17}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6261, 101, 156,
12, 95, 218, 212, 209} \begin {gather*} \frac {17}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{24 x}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{12 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 218
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{-\frac {3}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1-a x)^{3/4}}{x^4 (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {1}{3} \int \frac {-\frac {7 a}{2}+2 a^2 x}{x^3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {1}{6} \int \frac {-\frac {23 a^2}{4}+\frac {7 a^3 x}{2}}{x^2 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac {1}{6} \int -\frac {51 a^3}{8 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}-\frac {1}{16} \left (17 a^3\right ) \int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}-\frac {1}{4} \left (17 a^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac {1}{8} \left (17 a^3\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}{8} \left (17 a^3\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-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac {17}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {17}{8} a^3 \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 = 0.03, size = 78, normalized size = 0.56 \begin {gather*} \frac {(1-a x)^{3/4} \left (-8+6 a x-9 a^2 x^2-23 a^3 x^3+34 a^3 x^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{1+a x}\right )\right )}{24 x^3 (1+a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{4}}\, dx\]
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 = 153, normalized size = 1.10 \begin {gather*} \frac {102 \, a^{3} x^{3} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, {\left (23 \, a^{3} x^{3} - 37 \, a^{2} x^{2} + 22 \, a x - 8\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}} \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^{4} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, 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^4\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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