Integrand size = 27, antiderivative size = 58 \[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=-\frac {15}{4} \arctan \left (\frac {2 \sqrt {-x+x^3}}{-1-2 x+x^2}\right )-\frac {13}{4} \text {arctanh}\left (\frac {-\frac {1}{2}+x+\frac {x^2}{2}}{\sqrt {-x+x^3}}\right ) \] Output:
-15/4*arctan(2*(x^3-x)^(1/2)/(x^2-2*x-1))-13/4*arctanh((-1/2+x+1/2*x^2)/(x ^3-x)^(1/2))
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.45 \[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \left ((-1+14 i) \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )+(14-i) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^2}}{\sqrt {x}}\right )\right )}{\sqrt {x \left (-1+x^2\right )}} \] Input:
Integrate[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]
Output:
((1/4 + I/4)*Sqrt[x]*Sqrt[-1 + x^2]*((-1 + 14*I)*ArcTan[((1 + I)*Sqrt[x])/ Sqrt[-1 + x^2]] + (14 - I)*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^2])/Sqrt[x]]))/ Sqrt[x*(-1 + x^2)]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2467, 25, 2035, 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.
\(\displaystyle \int \frac {7 x^2+x-7}{\left (x^2+1\right ) \sqrt {x^3-x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int -\frac {-7 x^2-x+7}{\sqrt {x} \sqrt {x^2-1} \left (x^2+1\right )}dx}{\sqrt {x^3-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-1} \int \frac {-7 x^2-x+7}{\sqrt {x} \sqrt {x^2-1} \left (x^2+1\right )}dx}{\sqrt {x^3-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {-7 x^2-x+7}{\sqrt {x^2-1} \left (x^2+1\right )}d\sqrt {x}}{\sqrt {x^3-x}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \int \left (\frac {14-x}{\sqrt {x^2-1} \left (x^2+1\right )}-\frac {7}{\sqrt {x^2-1}}\right )d\sqrt {x}}{\sqrt {x^3-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \left (\left (\frac {15}{8}-\frac {13 i}{8}\right ) \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )+\left (\frac {13}{8}-\frac {15 i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )\right )}{\sqrt {x^3-x}}\) |
Input:
Int[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]
Output:
(-2*Sqrt[x]*Sqrt[-1 + x^2]*((15/8 - (13*I)/8)*ArcTan[((1 + I)*Sqrt[x])/Sqr t[-1 + x^2]] + (13/8 - (15*I)/8)*ArcTanh[((1 + I)*Sqrt[x])/Sqrt[-1 + x^2]] ))/Sqrt[-x + x^3]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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 = 2.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.59
method | result | size |
default | \(\frac {13 \ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {15 \arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}-\frac {13 \ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {15 \arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}\) | \(92\) |
pseudoelliptic | \(\frac {13 \ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {15 \arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}-\frac {13 \ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {15 \arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}\) | \(92\) |
elliptic | \(-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {7 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}\) | \(211\) |
trager | \(-\frac {13 \ln \left (\frac {464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}-696 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x +4141 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}-464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}+6000 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-774 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x +8051 x^{2}-4141 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+13875 \sqrt {x^{3}-x}+3486 x -8051}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-17 x -41\right )}^{2}}\right )}{4}-\frac {\ln \left (\frac {464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}-696 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x +4141 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}-464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}+6000 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-774 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x +8051 x^{2}-4141 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+13875 \sqrt {x^{3}-x}+3486 x -8051}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-17 x -41\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) \ln \left (\frac {928 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}-1392 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x +3782 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}-928 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}-12000 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-16548 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x +1477 x^{2}-3782 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-50250 \sqrt {x^{3}-x}-41778 x -1477}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+43 x -11\right )}^{2}}\right )}{2}\) | \(543\) |
Input:
int((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x,method=_RETURNVERBOSE)
Output:
13/8*ln((x^2-2*(x^3-x)^(1/2)+2*x-1)/x)+15/4*arctan(((x^3-x)^(1/2)-x)/x)-13 /8*ln((x^2+2*(x^3-x)^(1/2)+2*x-1)/x)+15/4*arctan(((x^3-x)^(1/2)+x)/x)
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.31 \[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {15}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + \frac {13}{8} \, \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) \] Input:
integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="fricas")
Output:
15/4*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^3 - x)) + 13/8*log((x^4 + 8*x^3 + 2 *x^2 - 4*sqrt(x^3 - x)*(x^2 + 2*x - 1) - 8*x + 1)/(x^4 + 2*x^2 + 1))
\[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\int \frac {7 x^{2} + x - 7}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:
integrate((7*x**2+x-7)/(x**2+1)/(x**3-x)**(1/2),x)
Output:
Integral((7*x**2 + x - 7)/(sqrt(x*(x - 1)*(x + 1))*(x**2 + 1)), x)
\[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {7 \, x^{2} + x - 7}{\sqrt {x^{3} - x} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="maxima")
Output:
integrate((7*x^2 + x - 7)/(sqrt(x^3 - x)*(x^2 + 1)), x)
\[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {7 \, x^{2} + x - 7}{\sqrt {x^{3} - x} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="giac")
Output:
integrate((7*x^2 + x - 7)/(sqrt(x^3 - x)*(x^2 + 1)), x)
Time = 8.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.72 \[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {-14\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )+\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,\left (14+1{}\mathrm {i}\right )+\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (1{}\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,\left (14-\mathrm {i}\right )}{\sqrt {x^3-x}} \] Input:
int((x + 7*x^2 - 7)/((x^3 - x)^(1/2)*(x^2 + 1)),x)
Output:
((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(-1i, asin((-x)^(1/2)), -1)*(14 + 1i) - 14*(-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF(asin(( -x)^(1/2)), -1) + (-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(1i, as in((-x)^(1/2)), -1)*(14 - 1i))/(x^3 - x)^(1/2)
\[ \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx=7 \left (\int \frac {\sqrt {x}\, \sqrt {x^{2}-1}\, x}{x^{4}-1}d x \right )-7 \left (\int \frac {\sqrt {x}\, \sqrt {x^{2}-1}}{x^{5}-x}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{2}-1}}{x^{4}-1}d x \] Input:
int((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x)
Output:
7*int((sqrt(x)*sqrt(x**2 - 1)*x)/(x**4 - 1),x) - 7*int((sqrt(x)*sqrt(x**2 - 1))/(x**5 - x),x) + int((sqrt(x)*sqrt(x**2 - 1))/(x**4 - 1),x)