Integrand size = 30, antiderivative size = 53 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {1+x^3}}{3}+\frac {4}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]
(2*Sqrt[1 + x^3])/3 + (4*ArcTanh[Sqrt[1 + x^3]])/3 - (5*Sqrt[2]*ArcTanh[Sq rt[1 + x^3]/Sqrt[2]])/3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.41 (sec) , antiderivative size = 1215, normalized size of antiderivative = 22.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1388, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {x^3+1} \left (x^6+2 x^3+2\right )}{x \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {x^6+2 x^3+2}{x \left (x^3-1\right ) \sqrt {x^3+1}}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5}{3 (x-1) \sqrt {x^3+1}}-\frac {2}{\sqrt {x^3+1} x}+\frac {x^2}{\sqrt {x^3+1}}+\frac {5 (2 x+1)}{3 \left (x^2+x+1\right ) \sqrt {x^3+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \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}\) |
(2*Sqrt[1 + x^3])/3 + (((5*I)/3)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[((3 - 6*I) - (2 - 3*I)*Sqrt[3])/((4 + 6*I) - (2 + 4*I)*Sqrt[3])]*Sqrt[(1 - x + x^2)/ (1 + Sqrt[3] + x)^2])])/(Sqrt[2]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (10*Sqrt[((6 - 3*I) - (3 - 2*I)*Sqrt[3])/((-6 - 4*I) + (4 + 2*I) *Sqrt[3])]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(3^(1/4) *Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[((6 - 3*I) - (3 - 2*I)*Sqrt[3])/ ((-6 - 4*I) + (4 + 2*I)*Sqrt[3])]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]) ])/(3^(3/4)*(3*I - Sqrt[3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3 ]) - (5*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTanh[Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]/(Sqrt[2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2])] )/(3*Sqrt[2]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (4*ArcTanh [Sqrt[1 + x^3]])/3 - (10*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*E llipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3* 3^(1/4)*Sqrt[2 + Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (20*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*E llipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3* 3^(1/4)*(1 + (2 - I)*Sqrt[3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x ^3]) - (20*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^ 2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3...
3.7.81.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
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 | \(\text {Expression too large to display}\) | \(905\) |
2/3*(x^3+1)^(1/2)+2/3*ln((x^3+1)^(1/2)+1)-2/3*ln((x^3+1)^(1/2)-1)-5/3*2^(1 /2)*arctanh(1/2*(x^3+1)^(1/2)*2^(1/2))
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {5}{6} \, \sqrt {2} \log \left (\frac {x^{3} - 2 \, \sqrt {2} \sqrt {x^{3} + 1} + 3}{x^{3} - 1}\right ) + \frac {2}{3} \, \sqrt {x^{3} + 1} + \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
5/6*sqrt(2)*log((x^3 - 2*sqrt(2)*sqrt(x^3 + 1) + 3)/(x^3 - 1)) + 2/3*sqrt( x^3 + 1) + 2/3*log(sqrt(x^3 + 1) + 1) - 2/3*log(sqrt(x^3 + 1) - 1)
Time = 7.98 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {2 \sqrt {x^{3} + 1}}{3} + \frac {5 \sqrt {2} \left (\log {\left (\sqrt {x^{3} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {x^{3} + 1} + \sqrt {2} \right )}\right )}{6} - \frac {2 \log {\left (\sqrt {x^{3} + 1} - 1 \right )}}{3} + \frac {2 \log {\left (\sqrt {x^{3} + 1} + 1 \right )}}{3} \]
2*sqrt(x**3 + 1)/3 + 5*sqrt(2)*(log(sqrt(x**3 + 1) - sqrt(2)) - log(sqrt(x **3 + 1) + sqrt(2)))/6 - 2*log(sqrt(x**3 + 1) - 1)/3 + 2*log(sqrt(x**3 + 1 ) + 1)/3
\[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 2\right )} \sqrt {x^{3} + 1}}{{\left (x^{6} - 1\right )} x} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\frac {5}{6} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{3} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{3} + 1}\right )}}\right ) + \frac {2}{3} \, \sqrt {x^{3} + 1} + \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {2}{3} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \]
5/6*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(x^3 + 1))/(sqrt(2) + sqrt(x^3 + 1))) + 2/3*sqrt(x^3 + 1) + 2/3*log(sqrt(x^3 + 1) + 1) - 2/3*log(abs(sqrt (x^3 + 1) - 1))
Time = 0.05 (sec) , antiderivative size = 775, normalized size of antiderivative = 14.62 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
((2*x^3)/3 + 2/3)/(x^3 + 1)^(1/2) + (4*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/ 2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3 /2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellip ticPi((3^(1/2)*1i)/2 + 3/2, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i) /2 + 1/2))^(1/2) - (5*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/( (3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^( 1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i )/4 + 3/4, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1 /2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/ 2)) - (2*((3^(1/2)*1i)/2 + 3/2)*(3*((3^(1/2)*1i)/2 - 1/2)^3 + 2)*((x + (3^ (1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*el lipticPi(((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 + 1/2), asin(((x + 1)/((3^ (1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2) ))/(((3^(1/2)*1i)/2 + 1/2)*(4*((3^(1/2)*1i)/2 - 1/2)^3 - 1)*(x^3 - x*(((3^ (1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*(( 3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 + 3/2)*(3*((3^(1/2)*1...