Integrand size = 25, antiderivative size = 93 \[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )}{3^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )}{3^{3/4}} \] Output:
-1/3*2^(1/2)*arctan((3+2*3^(1/2))^(1/2)*(x^3+1)^(1/2)/(x^2-x+1))*3^(1/4)-1 /3*2^(1/2)*arctanh((-3+2*3^(1/2))^(1/2)*(x^3+1)^(1/2)/(x^2-x+1))*3^(1/4)
Time = 1.51 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )+\text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )\right )}{3^{3/4}} \] Input:
Integrate[(2 + x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]
Output:
-((Sqrt[2]*(ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)] + Ar cTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]))/3^(3/4))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.02 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7279, 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.
\(\displaystyle \int \frac {x^2+2}{\left (x^2+2 x-2\right ) \sqrt {x^3+1}} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {1}{\sqrt {x^3+1}}+\frac {2 (2-x)}{\left (x^2+2 x-2\right ) \sqrt {x^3+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {2 \left (7+4 \sqrt {3}\right )} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )}{3^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{3^{3/4}}\) |
Input:
Int[(2 + x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]
Output:
-((Sqrt[2]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/3^(3/4)) - (Sqrt[2]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/3^(3/4) - (Sqrt[2]*(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^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqr t[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x) ^2]*Sqrt[1 + x^3]) - (Sqrt[2*(7 + 4*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^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3] )
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 1.34 (sec) , antiderivative size = 1500, normalized size of antiderivative = 16.13
method | result | size |
default | \(\text {Expression too large to display}\) | \(1500\) |
elliptic | \(\text {Expression too large to display}\) | \(1705\) |
Input:
int((x^2+2)/(x^2+2*x-2)/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(-1/2*I*3^(1/2)+3/2)*((1+x)/(-1/2*I*3^(1/2)+3/2))^(1/2)*((x-1/2-1/2*I*3^ (1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1 /2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((1+x)/(-1/2*I*3^(1/2)+3/2))^(1/2),((- 3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+I*(1/(-1/2*I*3^(1/2)+3/2)+ 1/(-1/2*I*3^(1/2)+3/2)*x)^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2-1/2*I* 3^(1/2))-1/2*I/(-3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(-3/2+1/2*I*3^(1/2)) *x-1/2/(-3/2+1/2*I*3^(1/2))+1/2*I/(-3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3 +1)^(1/2)*EllipticPi(((1+x)/(-1/2*I*3^(1/2)+3/2))^(1/2),-1/3*(-3/2+1/2*I*3 ^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2) -3*(1/(-1/2*I*3^(1/2)+3/2)+1/(-1/2*I*3^(1/2)+3/2)*x)^(1/2)*(1/(-3/2-1/2*I* 3^(1/2))*x-1/2/(-3/2-1/2*I*3^(1/2))-1/2*I/(-3/2-1/2*I*3^(1/2))*3^(1/2))^(1 /2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+1/2*I/(-3/2+1/2*I*3 ^(1/2))*3^(1/2))^(1/2)/(x^3+1)^(1/2)*EllipticPi(((1+x)/(-1/2*I*3^(1/2)+3/2 ))^(1/2),-1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2 *I*3^(1/2)))^(1/2))-I*(1/(-1/2*I*3^(1/2)+3/2)+1/(-1/2*I*3^(1/2)+3/2)*x)^(1 /2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/2/(-3/2-1/2*I*3^(1/2))-1/2*I/(-3/2-1/2*I*3 ^(1/2))*3^(1/2))^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/2/(-3/2+1/2*I*3^(1/2))+ 1/2*I/(-3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3+1)^(1/2)*EllipticPi(((1+x)/ (-1/2*I*3^(1/2)+3/2))^(1/2),-1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I *3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(1/(-1/2*I*3^(1/2)+3/2)+1/(-1/2*...
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (73) = 146\).
Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.28 \[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{2} \, \left (\frac {4}{27}\right )^{\frac {1}{4}} \arctan \left (\frac {3 \, {\left (3 \, \left (\frac {4}{27}\right )^{\frac {3}{4}} {\left (x^{2} + 2 \, x + 4\right )} - 2 \, \left (\frac {4}{27}\right )^{\frac {1}{4}} {\left (x^{2} - 2 \, x\right )}\right )}}{8 \, \sqrt {x^{3} + 1}}\right ) - \frac {1}{4} \, \left (\frac {4}{27}\right )^{\frac {1}{4}} \log \left (\frac {2 \, x^{4} - 4 \, x^{3} + 12 \, x^{2} + 24 \, \sqrt {\frac {1}{3}} {\left (x^{3} + 1\right )} + 3 \, \sqrt {x^{3} + 1} {\left (9 \, \left (\frac {4}{27}\right )^{\frac {3}{4}} {\left (x^{2} + 2\right )} - 2 \, \left (\frac {4}{27}\right )^{\frac {1}{4}} {\left (x^{2} - 4 \, x - 2\right )}\right )} + 8 \, x + 8}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{4} \, \left (\frac {4}{27}\right )^{\frac {1}{4}} \log \left (\frac {2 \, x^{4} - 4 \, x^{3} + 12 \, x^{2} + 24 \, \sqrt {\frac {1}{3}} {\left (x^{3} + 1\right )} - 3 \, \sqrt {x^{3} + 1} {\left (9 \, \left (\frac {4}{27}\right )^{\frac {3}{4}} {\left (x^{2} + 2\right )} - 2 \, \left (\frac {4}{27}\right )^{\frac {1}{4}} {\left (x^{2} - 4 \, x - 2\right )}\right )} + 8 \, x + 8}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \] Input:
integrate((x^2+2)/(x^2+2*x-2)/(x^3+1)^(1/2),x, algorithm="fricas")
Output:
-1/2*(4/27)^(1/4)*arctan(3/8*(3*(4/27)^(3/4)*(x^2 + 2*x + 4) - 2*(4/27)^(1 /4)*(x^2 - 2*x))/sqrt(x^3 + 1)) - 1/4*(4/27)^(1/4)*log((2*x^4 - 4*x^3 + 12 *x^2 + 24*sqrt(1/3)*(x^3 + 1) + 3*sqrt(x^3 + 1)*(9*(4/27)^(3/4)*(x^2 + 2) - 2*(4/27)^(1/4)*(x^2 - 4*x - 2)) + 8*x + 8)/(x^4 + 4*x^3 - 8*x + 4)) + 1/ 4*(4/27)^(1/4)*log((2*x^4 - 4*x^3 + 12*x^2 + 24*sqrt(1/3)*(x^3 + 1) - 3*sq rt(x^3 + 1)*(9*(4/27)^(3/4)*(x^2 + 2) - 2*(4/27)^(1/4)*(x^2 - 4*x - 2)) + 8*x + 8)/(x^4 + 4*x^3 - 8*x + 4))
\[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {x^{2} + 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{2} + 2 x - 2\right )}\, dx \] Input:
integrate((x**2+2)/(x**2+2*x-2)/(x**3+1)**(1/2),x)
Output:
Integral((x**2 + 2)/(sqrt((x + 1)*(x**2 - x + 1))*(x**2 + 2*x - 2)), x)
\[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \] Input:
integrate((x^2+2)/(x^2+2*x-2)/(x^3+1)^(1/2),x, algorithm="maxima")
Output:
integrate((x^2 + 2)/(sqrt(x^3 + 1)*(x^2 + 2*x - 2)), x)
\[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \] Input:
integrate((x^2+2)/(x^2+2*x-2)/(x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate((x^2 + 2)/(sqrt(x^3 + 1)*(x^2 + 2*x - 2)), x)
Time = 0.22 (sec) , antiderivative size = 509, normalized size of antiderivative = 5.47 \[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx =\text {Too large to display} \] Input:
int((x^2 + 2)/((x^3 + 1)^(1/2)*(2*x + x^2 - 2)),x)
Output:
(2*((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)*ellipticF(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) + ((2*3^(1/2) - 6)*((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)*((3^(1/2)*1i)/2 + 3/2))/3, 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) + 6)*((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)*((3^(1/2)*1i)/2 + 3/2))/3, 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))
\[ \int \frac {2+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=2 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )+\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \] Input:
int((x^2+2)/(x^2+2*x-2)/(x^3+1)^(1/2),x)
Output:
2*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x) + int((s qrt(x**3 + 1)*x**2)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x)