Optimal. Leaf size=42 \[ \frac {4}{3} \sqrt [4]{x^3+1}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 42, 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 = {266, 50, 63, 212, 206, 203} \begin {gather*} \frac {4}{3} \sqrt [4]{x^3+1}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x} \, dx,x,x^3\right )\\ &=\frac {4}{3} \sqrt [4]{1+x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^3\right )\\ &=\frac {4}{3} \sqrt [4]{1+x^3}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=\frac {4}{3} \sqrt [4]{1+x^3}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=\frac {4}{3} \sqrt [4]{1+x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {4}{3} \sqrt [4]{x^3+1}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {4}{3} \sqrt [4]{1+x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 44, normalized size = 1.05 \begin {gather*} \frac {4}{3} \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} - \frac {2}{3} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{3} \, \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]
________________________________________________________________________________________
giac [A] time = 0.33, size = 45, normalized size = 1.07 \begin {gather*} \frac {4}{3} \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} - \frac {2}{3} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{3} \, \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]
________________________________________________________________________________________
maple [C] time = 1.56, size = 45, normalized size = 1.07
method | result | size |
meijerg | \(-\frac {-4 \left (4-3 \ln \relax (2)+\frac {\pi }{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )-\hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right ) \Gamma \left (\frac {3}{4}\right ) x^{3}}{12 \Gamma \left (\frac {3}{4}\right )}\) | \(45\) |
trager | \(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{3}+\frac {\ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}}-x^{3}-2 \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}}-2}{x^{3}}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{3}+1\right )^{\frac {1}{4}}}{x^{3}}\right )}{3}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 44, normalized size = 1.05 \begin {gather*} \frac {4}{3} \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} - \frac {2}{3} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{3} \, \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]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 30, normalized size = 0.71 \begin {gather*} \frac {4\,{\left (x^3+1\right )}^{1/4}}{3}-\frac {2\,\mathrm {atanh}\left ({\left (x^3+1\right )}^{1/4}\right )}{3}-\frac {2\,\mathrm {atan}\left ({\left (x^3+1\right )}^{1/4}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.79, size = 37, normalized size = 0.88 \begin {gather*} - \frac {x^{\frac {3}{4}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________