Optimal. Leaf size=197 \[ -\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}-\frac {31}{128} a^5 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {31}{128} a^5 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {31}{128} a^5 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {31}{128} a^5 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{a x+1}}{1920 x}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}{960 x^2}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{48 x^3}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{40 x^4} \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 {1}{2} \tanh ^{-1}(a x)}}{x^6} \, dx &=\int \frac {\sqrt [4]{1+a x}}{x^6 \sqrt [4]{1-a x}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}+\frac {1}{5} \int \frac {\frac {9 a}{2}+4 a^2 x}{x^5 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {1}{20} \int \frac {-\frac {55 a^2}{4}-\frac {27 a^3 x}{2}}{x^4 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}+\frac {1}{60} \int \frac {\frac {269 a^3}{8}+\frac {55 a^4 x}{2}}{x^3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {1}{120} \int \frac {-\frac {611 a^4}{16}-\frac {269 a^5 x}{8}}{x^2 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}+\frac {1}{120} \int \frac {465 a^5}{32 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}+\frac {1}{256} \left (31 a^5\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}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}+\frac {1}{64} \left (31 a^5\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}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}-\frac {1}{128} \left (31 a^5\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}{128} \left (31 a^5\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}}{5 x^5}-\frac {9 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{40 x^4}-\frac {11 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{48 x^3}-\frac {269 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{960 x^2}-\frac {611 a^4 (1-a x)^{3/4} \sqrt [4]{1+a x}}{1920 x}-\frac {31}{128} a^5 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {31}{128} a^5 \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 = 0.04, size = 94, normalized size = 0.48 \begin {gather*} -\frac {(1-a x)^{3/4} \left (384+816 a x+872 a^2 x^2+978 a^3 x^3+1149 a^4 x^4+611 a^5 x^5+310 a^5 x^5 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{1+a x}\right )\right )}{1920 x^5 (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 \[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{6}}\, dx\]
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 = 169, normalized size = 0.86 \begin {gather*} -\frac {930 \, a^{5} x^{5} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 465 \, a^{5} x^{5} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 465 \, a^{5} x^{5} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (611 \, a^{5} x^{5} - 73 \, a^{4} x^{4} - 98 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 48 \, a x - 384\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{3840 \, x^{5}} \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 {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{x^{6}}\, 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 {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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