Integrand size = 34, antiderivative size = 16 \[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=2 \arctan \left (x \sqrt {1-x^3}\right ) \] Output:
2*arctan(x*(-x^3+1)^(1/2))
Time = 2.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=2 \arctan \left (x \sqrt {1-x^3}\right ) \] Input:
Integrate[(2 - 5*x^3)/(Sqrt[1 - x^3]*(1 + x^2*(1 - x^3))),x]
Output:
2*ArcTan[x*Sqrt[1 - x^3]]
Time = 0.54 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7260, 216}
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.
\(\displaystyle \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (\left (1-x^3\right ) x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 7260 |
\(\displaystyle 2 \int \frac {1}{\left (1-x^3\right ) x^2+1}d\left (x \sqrt {1-x^3}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 2 \arctan \left (x \sqrt {1-x^3}\right )\) |
Input:
Int[(2 - 5*x^3)/(Sqrt[1 - x^3]*(1 + x^2*(1 - x^3))),x]
Output:
2*ArcTan[x*Sqrt[1 - x^3]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(u_)*(v_)^(r_.)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(q_.))^(m_.), x_Symbol] : > With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, Simp[c*(p/(r + 1)) Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r}, x] && EqQ[p, q*(r + 1)] && NeQ[r, -1] && Integ erQ[p/(r + 1)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.88
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {-x^{3}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{2}-1}\right )\) | \(62\) |
default | \(-i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \sqrt {2}\, \sqrt {i \left (1+2 x -i \sqrt {3}\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+i \sqrt {3}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right )\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )\) | \(191\) |
elliptic | \(-i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \sqrt {2}\, \sqrt {i \left (1+2 x -i \sqrt {3}\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+i \sqrt {3}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right )\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )\) | \(191\) |
Input:
int((-5*x^3+2)/(-x^3+1)^(1/2)/(1+x^2*(-x^3+1)),x,method=_RETURNVERBOSE)
Output:
RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^5-RootOf(_Z^2+1)*x^2+2*x*(-x^3+1)^(1/2 )+RootOf(_Z^2+1))/(x^5-x^2-1))
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=\arctan \left (\frac {{\left (x^{5} - x^{2} + 1\right )} \sqrt {-x^{3} + 1}}{2 \, {\left (x^{4} - x\right )}}\right ) \] Input:
integrate((-5*x^3+2)/(-x^3+1)^(1/2)/(1+x^2*(-x^3+1)),x, algorithm="fricas" )
Output:
arctan(1/2*(x^5 - x^2 + 1)*sqrt(-x^3 + 1)/(x^4 - x))
\[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=\int \frac {5 x^{3} - 2}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{5} - x^{2} - 1\right )}\, dx \] Input:
integrate((-5*x**3+2)/(-x**3+1)**(1/2)/(1+x**2*(-x**3+1)),x)
Output:
Integral((5*x**3 - 2)/(sqrt(-(x - 1)*(x**2 + x + 1))*(x**5 - x**2 - 1)), x )
\[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=\int { \frac {5 \, x^{3} - 2}{{\left ({\left (x^{3} - 1\right )} x^{2} - 1\right )} \sqrt {-x^{3} + 1}} \,d x } \] Input:
integrate((-5*x^3+2)/(-x^3+1)^(1/2)/(1+x^2*(-x^3+1)),x, algorithm="maxima" )
Output:
integrate((5*x^3 - 2)/(((x^3 - 1)*x^2 - 1)*sqrt(-x^3 + 1)), x)
\[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=\int { \frac {5 \, x^{3} - 2}{{\left ({\left (x^{3} - 1\right )} x^{2} - 1\right )} \sqrt {-x^{3} + 1}} \,d x } \] Input:
integrate((-5*x^3+2)/(-x^3+1)^(1/2)/(1+x^2*(-x^3+1)),x, algorithm="giac")
Output:
integrate((5*x^3 - 2)/(((x^3 - 1)*x^2 - 1)*sqrt(-x^3 + 1)), x)
Time = 23.73 (sec) , antiderivative size = 170, normalized size of antiderivative = 10.62 \[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=\sum _{k=1}^5\frac {\sqrt {6}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x-1\right )}\,\Pi \left (-\frac {3+\sqrt {3}\,1{}\mathrm {i}}{2\,\left (\mathrm {root}\left (z^5-z^2-1,z,k\right )-1\right )};\mathrm {asin}\left (\frac {\sqrt {6}\,\sqrt {\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x-1\right )}}{6}\right )\middle |\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {3\,x+3-\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {3}\,1{}\mathrm {i}}\,\sqrt {3\,x+3+\sqrt {3}\,x\,1{}\mathrm {i}-\sqrt {3}\,1{}\mathrm {i}}}{18\,\left (\mathrm {root}\left (z^5-z^2-1,z,k\right )-1\right )\,\sqrt {1-x^3}\,\mathrm {root}\left (z^5-z^2-1,z,k\right )} \] Input:
int((5*x^3 - 2)/((1 - x^3)^(1/2)*(x^2*(x^3 - 1) - 1)),x)
Output:
symsum((6^(1/2)*((3^(1/2)*1i)/2 + 3/2)*((3^(1/2)*1i - 3)*(x - 1))^(1/2)*el lipticPi(-(3^(1/2)*1i + 3)/(2*(root(z^5 - z^2 - 1, z, k) - 1)), asin((6^(1 /2)*((3^(1/2)*1i - 3)*(x - 1))^(1/2))/6), (3^(1/2)*1i)/2 + 1/2)*(3*x - 3^( 1/2)*x*1i + 3^(1/2)*1i + 3)^(1/2)*(3*x + 3^(1/2)*x*1i - 3^(1/2)*1i + 3)^(1 /2))/(18*(root(z^5 - z^2 - 1, z, k) - 1)*(1 - x^3)^(1/2)*root(z^5 - z^2 - 1, z, k)), k, 1, 5)
\[ \int \frac {2-5 x^3}{\sqrt {1-x^3} \left (1+x^2 \left (1-x^3\right )\right )} \, dx=2 \left (\int \frac {\sqrt {-x^{3}+1}}{x^{8}-2 x^{5}-x^{3}+x^{2}+1}d x \right )-5 \left (\int \frac {\sqrt {-x^{3}+1}\, x^{3}}{x^{8}-2 x^{5}-x^{3}+x^{2}+1}d x \right ) \] Input:
int((-5*x^3+2)/(-x^3+1)^(1/2)/(1+x^2*(-x^3+1)),x)
Output:
2*int(sqrt( - x**3 + 1)/(x**8 - 2*x**5 - x**3 + x**2 + 1),x) - 5*int((sqrt ( - x**3 + 1)*x**3)/(x**8 - 2*x**5 - x**3 + x**2 + 1),x)