Optimal. Leaf size=39 \[ -\frac {2 \sqrt [4]{1+x^2}}{x}-\text {ArcTan}\left (\sqrt [4]{1+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+x^2}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1821, 272, 65,
218, 212, 209} \begin {gather*} -\text {ArcTan}\left (\sqrt [4]{x^2+1}\right )-\frac {2 \sqrt [4]{x^2+1}}{x}-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rule 1821
Rubi steps
\begin {align*} \int \frac {2+x+x^2}{x^2 \left (1+x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{1+x^2}}{x}+\int \frac {1}{x \left (1+x^2\right )^{3/4}} \, dx\\ &=-\frac {2 \sqrt [4]{1+x^2}}{x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {2 \sqrt [4]{1+x^2}}{x}+2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=-\frac {2 \sqrt [4]{1+x^2}}{x}-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^2}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=-\frac {2 \sqrt [4]{1+x^2}}{x}-\tan ^{-1}\left (\sqrt [4]{1+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 10.01, size = 39, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{1+x^2}}{x}-\text {ArcTan}\left (\sqrt [4]{1+x^2}\right )-\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 = 1.32, size = 56, normalized size = 1.44
method | result | size |
risch | \(-\frac {2 \left (x^{2}+1\right )^{\frac {1}{4}}}{x}+\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{2 \Gamma \left (\frac {3}{4}\right )}\) | \(56\) |
meijerg | \(x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )+\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{2 \Gamma \left (\frac {3}{4}\right )}-\frac {2 \hypergeom \left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], -x^{2}\right )}{x}\) | \(73\) |
trager | \(-\frac {2 \left (x^{2}+1\right )^{\frac {1}{4}}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{2}+1\right )^{\frac {1}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}+\frac {\ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}+1}-x^{2}+2 \left (x^{2}+1\right )^{\frac {1}{4}}-2}{x^{2}}\right )}{2}\) | \(123\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (33) = 66\).
time = 0.80, size = 77, normalized size = 1.97 \begin {gather*} \frac {x \arctan \left (\frac {2 \, {\left ({\left (x^{2} + 1\right )}^{\frac {3}{4}} + {\left (x^{2} + 1\right )}^{\frac {1}{4}}\right )}}{x^{2}}\right ) + x \log \left (\frac {x^{2} - 2 \, {\left (x^{2} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {x^{2} + 1} - 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} + 2}{x^{2}}\right ) - 4 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{2 \, x} \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 = 1.84, size = 68, normalized size = 1.74 \begin {gather*} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )} - \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{x} - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {3}{2}} \Gamma \left (\frac {7}{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 [B]
time = 0.55, size = 48, normalized size = 1.23 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {3}{2};\ -x^2\right )-\mathrm {atanh}\left ({\left (x^2+1\right )}^{1/4}\right )-\mathrm {atan}\left ({\left (x^2+1\right )}^{1/4}\right )-\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{4};\ \frac {1}{2};\ -x^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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