Optimal. Leaf size=168 \[ \frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {11}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {83 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{192 x}-\frac {29 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{96 x^2}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6126, 99, 151, 12, 93, 298, 203, 206} \[ -\frac {29 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{192 x}+\frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {11}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x^3}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 298
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{-\frac {1}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac {\sqrt [4]{1-a x}}{x^5 \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {1}{4} \int \frac {-\frac {7 a}{2}+3 a^2 x}{x^4 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {1}{12} \int \frac {-\frac {29 a^2}{4}+7 a^3 x}{x^3 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {1}{24} \int \frac {-\frac {83 a^3}{8}+\frac {29 a^4 x}{4}}{x^2 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}-\frac {1}{24} \int -\frac {33 a^4}{16 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac {1}{128} \left (11 a^4\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac {1}{32} \left (11 a^4\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 {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}-\frac {1}{64} \left (11 a^4\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}{64} \left (11 a^4\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 {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x^3}-\frac {29 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{96 x^2}+\frac {83 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{192 x}+\frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {11}{64} a^4 \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 = 86, normalized size = 0.51 \[ \frac {\sqrt [4]{1-a x} \left (-66 a^4 x^4 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{a x+1}\right )+83 a^4 x^4+25 a^3 x^3-2 a^2 x^2+8 a x-48\right )}{192 x^4 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 165, normalized size = 0.98 \[ \frac {66 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 33 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, {\left (83 \, a^{3} x^{3} - 58 \, a^{2} x^{2} + 56 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \]
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 \[ \int \frac {1}{x^{5} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{5}}\, dx \]
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 \[ \int \frac {1}{x^{5} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {1}{x^5\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \]
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 \[ \int \frac {1}{x^{5} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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