Optimal. Leaf size=287 \[ \frac{181 x^3 \sqrt [4]{1-\frac{1}{a x}}}{240 a^2 \sqrt [4]{\frac{1}{a x}+1}}-\frac{1189 x^2 \sqrt [4]{1-\frac{1}{a x}}}{960 a^3 \sqrt [4]{\frac{1}{a x}+1}}+\frac{5533 x \sqrt [4]{1-\frac{1}{a x}}}{1920 a^4 \sqrt [4]{\frac{1}{a x}+1}}+\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{\frac{1}{a x}+1}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{x^5 \sqrt [4]{1-\frac{1}{a x}}}{5 \sqrt [4]{\frac{1}{a x}+1}}-\frac{21 x^4 \sqrt [4]{1-\frac{1}{a x}}}{40 a \sqrt [4]{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.162028, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac{181 x^3 \sqrt [4]{1-\frac{1}{a x}}}{240 a^2 \sqrt [4]{\frac{1}{a x}+1}}-\frac{1189 x^2 \sqrt [4]{1-\frac{1}{a x}}}{960 a^3 \sqrt [4]{\frac{1}{a x}+1}}+\frac{5533 x \sqrt [4]{1-\frac{1}{a x}}}{1920 a^4 \sqrt [4]{\frac{1}{a x}+1}}+\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{\frac{1}{a x}+1}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{x^5 \sqrt [4]{1-\frac{1}{a x}}}{5 \sqrt [4]{\frac{1}{a x}+1}}-\frac{21 x^4 \sqrt [4]{1-\frac{1}{a x}}}{40 a \sqrt [4]{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^4 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/4}}{x^6 \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{5} \operatorname{Subst}\left (\int \frac{\frac{21}{2 a}-\frac{10 x}{a^2}}{x^5 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{20} \operatorname{Subst}\left (\int \frac{\frac{181}{4 a^2}-\frac{42 x}{a^3}}{x^4 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{60} \operatorname{Subst}\left (\int \frac{\frac{1189}{8 a^3}-\frac{543 x}{4 a^4}}{x^3 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{120} \operatorname{Subst}\left (\int \frac{\frac{5533}{16 a^4}-\frac{1189 x}{4 a^5}}{x^2 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{120} \operatorname{Subst}\left (\int \frac{\frac{15045}{32 a^5}-\frac{5533 x}{16 a^6}}{x \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{60} a \operatorname{Subst}\left (\int \frac{15045}{64 a^6 x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1003 \operatorname{Subst}\left (\int \frac{1}{x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{256 a^5}\\ &=\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1003 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^5}\\ &=\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1003 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1003 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ &=\frac{26111 \sqrt [4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1-\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1189 \sqrt [4]{1-\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{181 \sqrt [4]{1-\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{21 \sqrt [4]{1-\frac{1}{a x}} x^4}{40 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^5}{5 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ \end{align*}
Mathematica [A] time = 5.37064, size = 198, normalized size = 0.69 \[ \frac{8 e^{-\frac{1}{2} \coth ^{-1}(a x)}-\frac{4117 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{192 \left (e^{-2 \coth ^{-1}(a x)}-1\right )}-\frac{1661 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{48 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^2}-\frac{233 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{6 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^3}-\frac{122 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{5 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^4}-\frac{32 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{5 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^5}+\frac{1003}{256} \log \left (1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )-\frac{1003}{256} \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right )-\frac{1003}{128} \tan ^{-1}\left (e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )}{a^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.33, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5489, size = 377, normalized size = 1.31 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{4 \,{\left (20585 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{4}} - 49120 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} + 61130 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} - 33816 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 7365 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{5 \,{\left (a x - 1\right )} a^{6}}{a x + 1} - \frac{10 \,{\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac{10 \,{\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac{{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac{30090 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} + \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} - \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{6}} - \frac{30720 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58489, size = 346, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (384 \, a^{5} x^{5} - 1008 \, a^{4} x^{4} + 1448 \, a^{3} x^{3} - 2378 \, a^{2} x^{2} + 5533 \, a x + 26111\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 30090 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{3840 \, a^{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 [A] time = 1.21479, size = 343, normalized size = 1.2 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{30090 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} + \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} - \frac{15045 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{6}} - \frac{30720 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{6}} - \frac{4 \,{\left (\frac{33816 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \frac{61130 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac{49120 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac{20585 \,{\left (a x - 1\right )}^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{4}} - 7365 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{a^{6}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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