Optimal. Leaf size=45 \[ -\frac {\sqrt [4]{1+x^2}}{2 x^2}+\frac {3}{4} \text {ArcTan}\left (\sqrt [4]{1+x^2}\right )+\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{1+x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 65,
218, 212, 209} \begin {gather*} \frac {3}{4} \text {ArcTan}\left (\sqrt [4]{x^2+1}\right )-\frac {\sqrt [4]{x^2+1}}{2 x^2}+\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^2\right )^{3/4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{1+x^2}}{2 x^2}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{1+x^2}}{2 x^2}-\frac {3}{2} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=-\frac {\sqrt [4]{1+x^2}}{2 x^2}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=-\frac {\sqrt [4]{1+x^2}}{2 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{1+x^2}\right )+\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{1+x^2}}{2 x^2}+\frac {3}{4} \text {ArcTan}\left (\sqrt [4]{1+x^2}\right )+\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.33, size = 52, normalized size = 1.16
method | result | size |
meijerg | \(\frac {\frac {21 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], -x^{2}\right )}{32}-\frac {3 \left (\frac {1}{3}-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{4}-\frac {\Gamma \left (\frac {3}{4}\right )}{x^{2}}}{2 \Gamma \left (\frac {3}{4}\right )}\) | \(52\) |
risch | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{4}}}{2 x^{2}}-\frac {3 \left (-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )\right )}{8 \Gamma \left (\frac {3}{4}\right )}\) | \(56\) |
trager | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{4}}}{2 x^{2}}-\frac {3 \ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}+1}-x^{2}+2 \left (x^{2}+1\right )^{\frac {1}{4}}-2}{x^{2}}\right )}{8}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{2}+1\right )^{\frac {1}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{8}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{2} + 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} + \frac {3}{4} \, \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{8} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {3}{8} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 58, normalized size = 1.29 \begin {gather*} \frac {6 \, x^{2} \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) + 3 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) - 3 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) - 4 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{8 \, x^{2}} \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 = 0.64, size = 32, normalized size = 0.71 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{2} + 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} + \frac {3}{4} \, \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{8} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {3}{8} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 33, normalized size = 0.73 \begin {gather*} \frac {3\,\mathrm {atan}\left ({\left (x^2+1\right )}^{1/4}\right )}{4}+\frac {3\,\mathrm {atanh}\left ({\left (x^2+1\right )}^{1/4}\right )}{4}-\frac {{\left (x^2+1\right )}^{1/4}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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