Optimal. Leaf size=42 \[ -\frac {2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {126, 323, 342,
234} \begin {gather*} -\frac {2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 126
Rule 234
Rule 323
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx &=\int \frac {1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx\\ &=\frac {\left (\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2} \, dx}{e^2 \sqrt [4]{1-x^2}}\\ &=-\frac {\left (\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^2}} \, dx,x,\frac {1}{x}\right )}{e^2 \sqrt [4]{1-x^2}}\\ &=-\frac {2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 23, normalized size = 0.55 \begin {gather*} -\frac {2 x \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};x^2\right )}{(e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (1-x \right )^{\frac {1}{4}} \left (e x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {1}{4}}}\, dx\]
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 14.87, size = 87, normalized size = 2.07 \begin {gather*} - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{8}, \frac {11}{8}, 1 & 1, \frac {5}{4}, \frac {3}{2} \\\frac {1}{2}, \frac {7}{8}, 1, \frac {11}{8}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi e^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{4}, \frac {3}{8}, \frac {3}{4}, \frac {7}{8}, \frac {5}{4}, 1 & \\\frac {3}{8}, \frac {7}{8} & \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi e^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \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 = -0.02 \begin {gather*} \int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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