Optimal. Leaf size=179 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {11 x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{24 a^2}+\frac {1}{3} x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {5 x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{12 a} \]
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Rubi [A] time = 0.09, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6171, 99, 151, 12, 93, 298, 203, 206} \[ \frac {11 x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{24 a^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {1}{3} x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {5 x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{12 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 298
Rule 6171
Rubi steps
\begin {align*} \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{x^4 \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-\frac {5}{2 a}+\frac {2 x}{a^2}}{x^3 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-\frac {11}{4 a^2}+\frac {5 x}{2 a^3}}{x^2 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{24 a^2}-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3-\frac {1}{6} \operatorname {Subst}\left (\int -\frac {9}{8 a^3 x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{24 a^2}-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3+\frac {3 \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 )}{16 a^3}\\ &=\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{24 a^2}-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3+\frac {3 \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 )}{4 a^3}\\ &=\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{24 a^2}-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3-\frac {3 \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 )}{8 a^3}+\frac {3 \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 )}{8 a^3}\\ &=\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{24 a^2}-\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{12 a}+\frac {1}{3} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}\\ \end {align*}
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Mathematica [C] time = 9.03, size = 389, normalized size = 2.17 \[ -\frac {e^{-\frac {5}{2} \coth ^{-1}(a x)} \left (256 e^{6 \coth ^{-1}(a x)} \left (1090 e^{2 \coth ^{-1}(a x)}+437 e^{4 \coth ^{-1}(a x)}+685\right ) \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {19}{4};e^{2 \coth ^{-1}(a x)}\right )+2048 e^{6 \coth ^{-1}(a x)} \left (38 e^{2 \coth ^{-1}(a x)}+17 e^{4 \coth ^{-1}(a x)}+21\right ) \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {19}{4};e^{2 \coth ^{-1}(a x)}\right )+4096 e^{6 \coth ^{-1}(a x)} \, _6F_5\left (\frac {7}{4},2,2,2,2,2;1,1,1,1,\frac {19}{4};e^{2 \coth ^{-1}(a x)}\right )+8192 e^{8 \coth ^{-1}(a x)} \, _6F_5\left (\frac {7}{4},2,2,2,2,2;1,1,1,1,\frac {19}{4};e^{2 \coth ^{-1}(a x)}\right )+4096 e^{10 \coth ^{-1}(a x)} \, _6F_5\left (\frac {7}{4},2,2,2,2,2;1,1,1,1,\frac {19}{4};e^{2 \coth ^{-1}(a x)}\right )+17244920 e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )-9077530 e^{4 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )-7043960 e^{6 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )+446985 e^{8 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )+22034705 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )-26688365 e^{2 \coth ^{-1}(a x)}-3731255 e^{4 \coth ^{-1}(a x)}+3122405 e^{6 \coth ^{-1}(a x)}-22034705\right )}{221760 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 102, normalized size = 0.57 \[ \frac {2 \, {\left (8 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 11\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{48 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 172, normalized size = 0.96 \[ -\frac {1}{48} \, a {\left (\frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} + \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} - \frac {9 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{4}} - \frac {4 \, {\left (\frac {6 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {29 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} - 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 187, normalized size = 1.04 \[ -\frac {1}{48} \, a {\left (\frac {4 \, {\left (29 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 6 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} + \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} - \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 157, normalized size = 0.88 \[ \frac {\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{4}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}+\frac {29\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{12}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {3\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3}-\frac {3\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3} \]
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 \[ \int x^{2} \sqrt [4]{\frac {a x - 1}{a x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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