Optimal. Leaf size=165 \[ -\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {55}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {55}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6261, 100,
156, 160, 12, 95, 304, 209, 212} \begin {gather*} -\frac {55}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{a x+1}}+\frac {55}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{a x+1}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{a x+1}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{a x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 160
Rule 209
Rule 212
Rule 304
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{-\frac {5}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1-a x)^{5/4}}{x^4 (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}-\frac {1}{3} \int \frac {\frac {13 a}{2}-6 a^2 x}{x^3 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}+\frac {1}{6} \int \frac {\frac {61 a^2}{4}-13 a^3 x}{x^2 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {1}{6} \int \frac {\frac {165 a^3}{8}-\frac {61 a^4 x}{4}}{x (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {\int \frac {165 a^4}{16 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{3 a}\\ &=-\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {1}{16} \left (55 a^3\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {1}{4} \left (55 a^3\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 {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}+\frac {1}{8} \left (55 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 (55 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 {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac {55}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {55}{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 = 0.02, size = 78, normalized size = 0.47 \begin {gather*} \frac {\sqrt [4]{1-a x} \left (-8+26 a x-61 a^2 x^2-287 a^3 x^3+330 a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{1+a x}\right )\right )}{24 x^3 \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 \[\int \frac {1}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{4}}\, dx\]
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 = 0.35, size = 200, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 330 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 165 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 165 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{48 \, {\left (a x^{4} + x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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