Optimal. Leaf size=197 \[ -\frac{269 a^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}{960 x^2}-\frac{11 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{48 x^3}-\frac{611 a^4 (1-a x)^{3/4} \sqrt [4]{a x+1}}{1920 x}-\frac{31}{128} a^5 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{31}{128} a^5 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{9 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{40 x^4}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{5 x^5} \]
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Rubi [A] time = 0.0955458, antiderivative size = 197, normalized size of antiderivative = 1., 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 = {6126, 99, 151, 12, 93, 212, 206, 203} \[ -\frac{269 a^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}{960 x^2}-\frac{11 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{48 x^3}-\frac{611 a^4 (1-a x)^{3/4} \sqrt [4]{a x+1}}{1920 x}-\frac{31}{128} a^5 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{31}{128} a^5 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{9 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{40 x^4}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
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 ) \operatorname{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 ) \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}{128} \left (31 a^5\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-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*}
Mathematica [C] time = 0.0295945, size = 94, normalized size = 0.48 \[ -\frac{(1-a x)^{3/4} \left (310 a^5 x^5 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+611 a^5 x^5+1149 a^4 x^4+978 a^3 x^3+872 a^2 x^2+816 a x+384\right )}{1920 x^5 (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77041, size = 400, normalized size = 2.03 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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