Optimal. Leaf size=31 \[ -\frac {\sqrt {x^3+1}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 47, 63, 207} \begin {gather*} -\frac {\sqrt {x^3+1}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{3 x^3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {\sqrt {1+x^3}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 44, normalized size = 1.42 \begin {gather*} -\frac {1}{3 x^3 \sqrt {x^3+1}}-\frac {1}{3 \sqrt {x^3+1}}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^3}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 44, normalized size = 1.42 \begin {gather*} -\frac {x^{3} \log \left (\sqrt {x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt {x^{3} + 1} - 1\right ) + 2 \, \sqrt {x^{3} + 1}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 38, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {1}{6} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{6} \, \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 = 0.21, size = 24, normalized size = 0.77
method | result | size |
default | \(-\frac {\sqrt {x^{3}+1}}{3 x^{3}}-\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{3}\) | \(24\) |
risch | \(-\frac {\sqrt {x^{3}+1}}{3 x^{3}}-\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{3}\) | \(24\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{3 x^{3}}-\frac {\arctanh \left (\sqrt {x^{3}+1}\right )}{3}\) | \(24\) |
trager | \(-\frac {\sqrt {x^{3}+1}}{3 x^{3}}+\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{6}\) | \(38\) |
meijerg | \(-\frac {\frac {2 \sqrt {\pi }}{x^{3}}-\left (-2 \ln \relax (2)-1+3 \ln \relax (x )\right ) \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{4 x^{3}}+\frac {2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{3}}+2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {1}{6} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{6} \, \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 = 0.06, size = 177, normalized size = 5.71 \begin {gather*} -\frac {\sqrt {x^3+1}}{3\,x^3}-\frac {\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 )}{\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 = 1.25, size = 27, normalized size = 0.87 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} - \frac {\sqrt {1 + \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________