Integrand size = 30, antiderivative size = 53 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.32 (sec) , antiderivative size = 1215, normalized size of antiderivative = 22.92, number of steps used = 33, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1600, 6857, 2161, 224, 2167, 2138, 551, 585, 95, 212, 272, 65, 213, 267, 6860, 211} \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {5 i (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \arctan \left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(3-6 i)-(2-3 i) \sqrt {3}}{(4+6 i)-(2+4 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {10 \sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \arctan \left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3^{3/4} \left (3 i-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {5 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \text {arctanh}\left (\frac {\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {2} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {4}{3} \text {arctanh}\left (\sqrt {x^3+1}\right )-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {10 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (97+56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {x^3+1}}{3} \]
[In]
[Out]
Rule 65
Rule 95
Rule 211
Rule 212
Rule 213
Rule 224
Rule 267
Rule 272
Rule 551
Rule 585
Rule 1600
Rule 2138
Rule 2161
Rule 2167
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+2 x^3+x^6}{x \left (-1+x^3\right ) \sqrt {1+x^3}} \, dx \\ & = \int \left (\frac {5}{3 (-1+x) \sqrt {1+x^3}}-\frac {2}{x \sqrt {1+x^3}}+\frac {x^2}{\sqrt {1+x^3}}+\frac {5 (1+2 x)}{3 \left (1+x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = \frac {5}{3} \int \frac {1}{(-1+x) \sqrt {1+x^3}} \, dx+\frac {5}{3} \int \frac {1+2 x}{\left (1+x+x^2\right ) \sqrt {1+x^3}} \, dx-2 \int \frac {1}{x \sqrt {1+x^3}} \, dx+\int \frac {x^2}{\sqrt {1+x^3}} \, dx \\ & = \frac {2 \sqrt {1+x^3}}{3}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+\frac {5}{3} \int \left (\frac {2}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {2}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx-\frac {5 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {5 \int \frac {1+\sqrt {3}+x}{(-1+x) \sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )} \\ & = \frac {2 \sqrt {1+x^3}}{3}-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )+\frac {10}{3} \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {10}{3} \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {\left (20 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (2-\sqrt {3}+\left (2+\sqrt {3}\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}-\frac {\left (20 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (20 \left (2-\sqrt {3}\right )^{3/2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (97+56 \sqrt {3},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (10 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]
[In]
[Out]
Time = 2.84 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {2 \sqrt {x^{3}+1}}{3}+\frac {2 \ln \left (\sqrt {x^{3}+1}+1\right )}{3}-\frac {2 \ln \left (\sqrt {x^{3}+1}-1\right )}{3}-\frac {5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}\right )}{3}\) | \(53\) |
pseudoelliptic | \(\frac {2 \sqrt {x^{3}+1}}{3}+\frac {2 \ln \left (\sqrt {x^{3}+1}+1\right )}{3}-\frac {2 \ln \left (\sqrt {x^{3}+1}-1\right )}{3}-\frac {5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}\right )}{3}\) | \(53\) |
trager | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{3}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+4 \sqrt {x^{3}+1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}{6}\) | \(88\) |
elliptic | \(\text {Expression too large to display}\) | \(905\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {5}{6} \, \sqrt {2} \log \left (\frac {x^{3} - 2 \, \sqrt {2} \sqrt {x^{3} + 1} + 3}{x^{3} - 1}\right ) + \frac {2}{3} \, \sqrt {x^{3} + 1} + \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
[In]
[Out]
Time = 7.98 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {x^{3} + 1}}{3} + \frac {5 \sqrt {2} \left (\log {\left (\sqrt {x^{3} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {x^{3} + 1} + \sqrt {2} \right )}\right )}{6} - \frac {2 \log {\left (\sqrt {x^{3} + 1} - 1 \right )}}{3} + \frac {2 \log {\left (\sqrt {x^{3} + 1} + 1 \right )}}{3} \]
[In]
[Out]
\[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 2\right )} \sqrt {x^{3} + 1}}{{\left (x^{6} - 1\right )} x} \,d x } \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {5}{6} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{3} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{3} + 1}\right )}}\right ) + \frac {2}{3} \, \sqrt {x^{3} + 1} + \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {2}{3} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 775, normalized size of antiderivative = 14.62 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
[In]
[Out]