Optimal. Leaf size=38 \[ \frac {7}{12} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1} \left (7 x^3+2\right )}{12 x^6} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {446, 78, 51, 63, 203} \begin {gather*} \frac {7 \sqrt {x^3-1}}{12 x^3}+\frac {7}{12} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 203
Rule 446
Rubi steps
\begin {align*} \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1+x}{\sqrt {-1+x} x^3} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \tan ^{-1}\left (\sqrt {-1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 50, normalized size = 1.32 \begin {gather*} \frac {1}{12} \sqrt {x^3-1} \left (\frac {7 \tanh ^{-1}\left (\sqrt {1-x^3}\right )}{\sqrt {1-x^3}}+\frac {7 x^3+2}{x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.06, size = 38, normalized size = 1.00 \begin {gather*} \frac {7}{12} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1} \left (7 x^3+2\right )}{12 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 34, normalized size = 0.89 \begin {gather*} \frac {7 \, x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (7 \, 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 = 0.31, size = 35, normalized size = 0.92 \begin {gather*} \frac {7 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{3} - 1}}{12 \, x^{6}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 36, normalized size = 0.95 \begin {gather*} \frac {7 \sqrt {x^{3}-1}}{12 x^{3}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}+\frac {\sqrt {x^{3}-1}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 60, normalized size = 1.58 \begin {gather*} \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 189, normalized size = 4.97 \begin {gather*} \frac {7\,\sqrt {x^3-1}}{12\,x^3}+\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {7\,\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+\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}}}\,\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 )}{4\,\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 [A] time = 45.16, size = 63, normalized size = 1.66 \begin {gather*} - \frac {1 - \frac {1}{x^{3} - 1}}{12 \left (1 + \frac {1}{x^{3} - 1}\right )^{2} \sqrt {x^{3} - 1}} - \frac {7 \operatorname {atan}{\left (\frac {1}{\sqrt {x^{3} - 1}} \right )}}{12} + \frac {2}{3 \left (1 + \frac {1}{x^{3} - 1}\right ) \sqrt {x^{3} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________