Optimal. Leaf size=52 \[ \frac {\left (1+x^3\right )^{3/4} \left (-4+5 x^3\right )}{24 x^6}+\frac {5}{48} \text {ArcTan}\left (\sqrt [4]{1+x^3}\right )-\frac {5}{48} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.17, number of steps
used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 65,
304, 209, 212} \begin {gather*} \frac {5}{48} \text {ArcTan}\left (\sqrt [4]{x^3+1}\right )+\frac {5 \left (x^3+1\right )^{3/4}}{24 x^3}-\frac {5}{48} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {\left (x^3+1\right )^{3/4}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps
\begin {align*} \int \frac {1}{x^7 \sqrt [4]{1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{6 x^6}-\frac {5}{24} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{6 x^6}+\frac {5 \left (1+x^3\right )^{3/4}}{24 x^3}+\frac {5}{96} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{6 x^6}+\frac {5 \left (1+x^3\right )^{3/4}}{24 x^3}+\frac {5}{24} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{6 x^6}+\frac {5 \left (1+x^3\right )^{3/4}}{24 x^3}-\frac {5}{48} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {5}{48} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{6 x^6}+\frac {5 \left (1+x^3\right )^{3/4}}{24 x^3}+\frac {5}{48} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {5}{48} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 52, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{3/4} \left (-4+5 x^3\right )}{24 x^6}+\frac {5}{48} \text {ArcTan}\left (\sqrt [4]{1+x^3}\right )-\frac {5}{48} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.86, size = 82, normalized size = 1.58
method | result | size |
risch | \(\frac {5 x^{6}+x^{3}-4}{24 x^{6} \left (x^{3}+1\right )^{\frac {1}{4}}}+\frac {5 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{3} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{192 \pi }\) | \(82\) |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {15 \pi \sqrt {2}\, x^{3} \hypergeom \left (\left [1, 1, \frac {13}{4}\right ], \left [2, 4\right ], -x^{3}\right )}{128 \Gamma \left (\frac {3}{4}\right )}+\frac {5 \left (\frac {33}{10}-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{32 \Gamma \left (\frac {3}{4}\right )}-\frac {\pi \sqrt {2}}{2 \Gamma \left (\frac {3}{4}\right ) x^{6}}+\frac {\pi \sqrt {2}}{4 \Gamma \left (\frac {3}{4}\right ) x^{3}}\right )}{6 \pi }\) | \(87\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {3}{4}} \left (5 x^{3}-4\right )}{24 x^{6}}+\frac {5 \ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}}-x^{3}-2 \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}}-2}{x^{3}}\right )}{96}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}-2 \left (x^{3}+1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{96}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.46, size = 74, normalized size = 1.42 \begin {gather*} -\frac {5 \, {\left (x^{3} + 1\right )}^{\frac {7}{4}} - 9 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{24 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {5}{48} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {5}{96} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {5}{96} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 65, normalized size = 1.25 \begin {gather*} \frac {10 \, x^{6} \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - 5 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + 5 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, {\left (5 \, x^{3} - 4\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{96 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.57, size = 32, normalized size = 0.62 \begin {gather*} - \frac {\Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{\frac {27}{4}} \Gamma \left (\frac {13}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 60, normalized size = 1.15 \begin {gather*} \frac {5 \, {\left (x^{3} + 1\right )}^{\frac {7}{4}} - 9 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{24 \, x^{6}} + \frac {5}{48} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {5}{96} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {5}{96} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.79, size = 45, normalized size = 0.87 \begin {gather*} \frac {5\,\mathrm {atan}\left ({\left (x^3+1\right )}^{1/4}\right )}{48}-\frac {5\,\mathrm {atanh}\left ({\left (x^3+1\right )}^{1/4}\right )}{48}-\frac {3\,{\left (x^3+1\right )}^{3/4}}{8\,x^6}+\frac {5\,{\left (x^3+1\right )}^{7/4}}{24\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________