Integrand size = 13, antiderivative size = 35 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 283, 221} \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=\frac {\text {arcsinh}\left (x^3\right )}{3}-\frac {\sqrt {x^6+1}}{3 x^3} \]
[In]
[Out]
Rule 221
Rule 281
Rule 283
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1+x^6}}{3 x^3}+\frac {\text {arcsinh}\left (x^3\right )}{3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \]
[In]
[Out]
Time = 1.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}+\frac {\operatorname {arcsinh}\left (x^{3}\right )}{3}\) | \(20\) |
pseudoelliptic | \(\frac {x^{3} \operatorname {arcsinh}\left (x^{3}\right )-\sqrt {x^{6}+1}}{3 x^{3}}\) | \(24\) |
trager | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+1}\right )}{3}\) | \(30\) |
meijerg | \(-\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{3}}-4 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{3}\right )}{12 \sqrt {\pi }}\) | \(31\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=-\frac {x^{3} \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) + x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \]
[In]
[Out]
Time = 0.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=- \frac {x^{3}}{3 \sqrt {x^{6} + 1}} + \frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {x^{6} + 1}} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=-\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=-\frac {2 \, \sqrt {\frac {1}{x^{6}} + 1} - \log \left (\sqrt {\frac {1}{x^{6}} + 1} + 1\right ) + \log \left (\sqrt {\frac {1}{x^{6}} + 1} - 1\right )}{6 \, \mathrm {sgn}\left (x\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {1+x^6}}{x^4} \, dx=\int \frac {\sqrt {x^6+1}}{x^4} \,d x \]
[In]
[Out]