Optimal. Leaf size=165 \[ -\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{a x+1}}-\frac {55}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {55}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{a x+1}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{a x+1}}+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{a x+1}} \]
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Rubi [A] time = 0.08, 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 = {6126, 98, 151, 155, 12, 93, 298, 203, 206} \[ -\frac {287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{a x+1}}-\frac {61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{a x+1}}-\frac {55}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {55}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{a x+1}}-\frac {\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{a x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 155
Rule 203
Rule 206
Rule 298
Rule 6126
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 ) \operatorname {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 ) \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}{8} \left (55 a^3\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 {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] time = 0.03, size = 78, normalized size = 0.47 \[ \frac {\sqrt [4]{1-a x} \left (330 a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{a x+1}\right )-287 a^3 x^3-61 a^2 x^2+26 a x-8\right )}{24 x^3 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 200, normalized size = 1.21 \[ -\frac {2 \, {\left (287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 330 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 165 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 165 \, {\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{48 \, {\left (a x^{4} + x^{3}\right )}} \]
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 \[ \int \frac {1}{x^{4} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, 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 \[ \int \frac {1}{x^{4} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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