Optimal. Leaf size=43 \[ \frac {\sqrt {x^3+1} \left (4 x^6-3 x^3-2\right )}{6 x^6}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.40, number of steps used = 12, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1821, 1612, 51, 63, 207} \begin {gather*} -\frac {\sqrt {x^3+1}}{2 x^3}+\frac {2 \sqrt {x^3+1}}{3}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{3 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 207
Rule 1612
Rule 1821
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \left (1+x^3+x^6\right )}{x^7 \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(2+x) \left (1+x+x^2\right )}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}+\frac {2}{x^3 \sqrt {1+x}}+\frac {3}{x^2 \sqrt {1+x}}+\frac {3}{x \sqrt {1+x}}\right ) \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{x^3}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-2 \tanh ^{-1}\left (\sqrt {1+x^3}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-\tanh ^{-1}\left (\sqrt {1+x^3}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 61, normalized size = 1.42 \begin {gather*} \frac {1}{3} \left (-4 \sqrt {x^3+1} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};x^3+1\right )+\frac {\sqrt {x^3+1} \left (2 x^3-3\right )}{x^3}-3 \tanh ^{-1}\left (\sqrt {x^3+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.05, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^3} \left (-2-3 x^3+4 x^6\right )}{6 x^6}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 57, normalized size = 1.33 \begin {gather*} -\frac {9 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 9 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (4 \, x^{6} - 3 \, x^{3} - 2\right )} \sqrt {x^{3} + 1}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 59, normalized size = 1.37 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{3} + 1}}{6 \, x^{6}} - \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {3}{4} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.25, size = 45, normalized size = 1.05
method | result | size |
default | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}-\frac {\sqrt {x^{3}+1}}{3 x^{6}}-\frac {\sqrt {x^{3}+1}}{2 x^{3}}\) | \(45\) |
risch | \(-\frac {3 x^{6}+5 x^{3}+2}{6 x^{6} \sqrt {x^{3}+1}}+\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}\) | \(45\) |
elliptic | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}-\frac {\sqrt {x^{3}+1}}{3 x^{6}}-\frac {\sqrt {x^{3}+1}}{2 x^{3}}\) | \(45\) |
trager | \(\frac {\sqrt {x^{3}+1}\, \left (4 x^{6}-3 x^{3}-2\right )}{6 x^{6}}-\frac {3 \ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{4}\) | \(48\) |
meijerg | \(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{3 \sqrt {\pi }}+\frac {\left (-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {\sqrt {\pi }}{x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{8 x^{3}}-\frac {\sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{3}}+\ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {\sqrt {\pi }}{3 x^{6}}+\frac {\sqrt {\pi }}{3 x^{3}}+\frac {\left (\frac {7}{6}-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{24 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{24 x^{6}}-\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{2}}{\sqrt {\pi }}\) | \(230\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.50, size = 85, normalized size = 1.98 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{6 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} - \frac {\sqrt {x^{3} + 1}}{x^{3}} - \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 198, normalized size = 4.60 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3}-\frac {\sqrt {x^3+1}}{2\,x^3}-\frac {\sqrt {x^3+1}}{3\,x^6}-\frac {9\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 151.83, size = 105, normalized size = 2.44 \begin {gather*} \frac {2 \sqrt {x^{3} + 1}}{3} + \frac {3 \log {\left (-1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{4} - \frac {3 \log {\left (1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{4} + \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} + \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )^{2}} + \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} - \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{3} + 1}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________