3.21.15 \(\int \frac {1}{(1+x) (-2+2 x+x^2-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac {i \sqrt {-x^4+x^2+2 x-2} \left (-\frac {1}{4} i \tanh ^{-1}\left (-\sqrt {x^2+2 x+2}+x+1\right )-\frac {7 i \tanh ^{-1}\left (\frac {\sqrt {x^2+2 x+2}}{\sqrt {5}}-\frac {x}{\sqrt {5}}+\frac {1}{\sqrt {5}}\right )}{20 \sqrt {5}}-\frac {i \left (x^3-2 x^2-2 x+4\right )}{20 (x-1)^2 \sqrt {x^2+2 x+2}}\right )}{(x-1) \sqrt {x^2+2 x+2}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((1 + x)*(-2 + 2*x + x^2 - x^4)^(3/2)),x]

[Out]

Defer[Int][1/((1 + x)*(-2 + 2*x + x^2 - x^4)^(3/2)), x]

Rubi steps

\begin {align*} \int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx &=\int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 121, normalized size = 0.85 \begin {gather*} \frac {7 \sqrt {5} \sqrt {x^2+2 x+2} (x-1)^2 \tanh ^{-1}\left (\frac {2 x+3}{\sqrt {5} \sqrt {x^2+2 x+2}}\right )-25 \sqrt {x^2+2 x+2} (x-1)^2 \tanh ^{-1}\left (\sqrt {(x+1)^2+1}\right )+10 \left (x^3-2 x^2-2 x+4\right )}{200 (x-1) \sqrt {-x^4+x^2+2 x-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x)*(-2 + 2*x + x^2 - x^4)^(3/2)),x]

[Out]

(10*(4 - 2*x - 2*x^2 + x^3) + 7*Sqrt[5]*(-1 + x)^2*Sqrt[2 + 2*x + x^2]*ArcTanh[(3 + 2*x)/(Sqrt[5]*Sqrt[2 + 2*x
+ x^2])] - 25*(-1 + x)^2*Sqrt[2 + 2*x + x^2]*ArcTanh[Sqrt[1 + (1 + x)^2]])/(200*(-1 + x)*Sqrt[-2 + 2*x + x^2
- x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 8.74, size = 143, normalized size = 1.00 \begin {gather*} -\frac {i \sqrt {-2+2 x+x^2-x^4} \left (-\frac {i \left (4-2 x-2 x^2+x^3\right )}{20 (-1+x)^2 \sqrt {2+2 x+x^2}}-\frac {1}{4} i \tanh ^{-1}\left (1+x-\sqrt {2+2 x+x^2}\right )-\frac {7 i \tanh ^{-1}\left (\frac {1}{\sqrt {5}}-\frac {x}{\sqrt {5}}+\frac {\sqrt {2+2 x+x^2}}{\sqrt {5}}\right )}{20 \sqrt {5}}\right )}{(-1+x) \sqrt {2+2 x+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 + x)*(-2 + 2*x + x^2 - x^4)^(3/2)),x]

[Out]

((-I)*Sqrt[-2 + 2*x + x^2 - x^4]*(((-1/20*I)*(4 - 2*x - 2*x^2 + x^3))/((-1 + x)^2*Sqrt[2 + 2*x + x^2]) - (I/4)
*ArcTanh[1 + x - Sqrt[2 + 2*x + x^2]] - (((7*I)/20)*ArcTanh[1/Sqrt[5] - x/Sqrt[5] + Sqrt[2 + 2*x + x^2]/Sqrt[5
]])/Sqrt[5]))/((-1 + x)*Sqrt[2 + 2*x + x^2])

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 174, normalized size = 1.22 \begin {gather*} -\frac {7 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-x^{4} + x^{2} + 2 \, x - 2} {\left (2 \, x + 3\right )}}{5 \, {\left (x^{3} + x^{2} - 2\right )}}\right ) - 25 \, {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\right )} \arctan \left (\frac {\sqrt {-x^{4} + x^{2} + 2 \, x - 2}}{x^{3} + x^{2} - 2}\right ) + 10 \, \sqrt {-x^{4} + x^{2} + 2 \, x - 2} {\left (x^{3} - 2 \, x^{2} - 2 \, x + 4\right )}}{200 \, {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(-x^4+x^2+2*x-2)^(3/2),x, algorithm="fricas")

[Out]

-1/200*(7*sqrt(5)*(x^5 - x^4 - x^3 - x^2 + 4*x - 2)*arctan(1/5*sqrt(5)*sqrt(-x^4 + x^2 + 2*x - 2)*(2*x + 3)/(x
^3 + x^2 - 2)) - 25*(x^5 - x^4 - x^3 - x^2 + 4*x - 2)*arctan(sqrt(-x^4 + x^2 + 2*x - 2)/(x^3 + x^2 - 2)) + 10*
sqrt(-x^4 + x^2 + 2*x - 2)*(x^3 - 2*x^2 - 2*x + 4))/(x^5 - x^4 - x^3 - x^2 + 4*x - 2)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(-x^4+x^2+2*x-2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.36, size = 118, normalized size = 0.83

method	result	size
risch	\(\frac {x^{3}-2 x^{2}-2 x +4}{20 \left (-1+x \right ) \sqrt {-\left (x^{2}+2 x +2\right ) \left (-1+x \right )^{2}}}-\frac {\left (\frac {7 \sqrt {5}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {5}}{10 \sqrt {-\left (-1+x \right )^{2}-1-4 x}}\right )}{200}+\frac {\arctan \left (\frac {1}{\sqrt {-\left (1+x \right )^{2}-1}}\right )}{8}\right ) \left (-1+x \right ) \sqrt {-x^{2}-2 x -2}}{\sqrt {-\left (x^{2}+2 x +2\right ) \left (-1+x \right )^{2}}}\)	\(118\)
trager	\(-\frac {\left (x^{3}-2 x^{2}-2 x +4\right ) \sqrt {-x^{4}+x^{2}+2 x -2}}{20 \left (-1+x \right )^{3} \left (x^{2}+2 x +2\right )}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+5\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+5\right ) x +5 \sqrt {-x^{4}+x^{2}+2 x -2}-3 \RootOf \left (\textit {\_Z}^{2}+5\right )}{\left (-1+x \right )^{2}}\right )}{200}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{4}+x^{2}+2 x -2}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right )}\right )}{8}\)	\(160\)
default	\(\frac {\left (-7 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, \sqrt {5}\, x^{2}+25 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, x^{2}+14 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, \sqrt {5}\, x -50 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, x -7 \sqrt {5}\, \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}-10 x^{3}+25 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}+20 x^{2}+20 x -40\right ) \left (-1+x \right ) \left (x^{2}+2 x +2\right )}{200 \left (-x^{4}+x^{2}+2 x -2\right )^{\frac {3}{2}}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x)/(-x^4+x^2+2*x-2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/20*(x^3-2*x^2-2*x+4)/(-1+x)/(-(x^2+2*x+2)*(-1+x)^2)^(1/2)-(7/200*5^(1/2)*arctan(1/10*(-6-4*x)*5^(1/2)/(-(-1+
x)^2-1-4*x)^(1/2))+1/8*arctan(1/(-(1+x)^2-1)^(1/2)))*(-1+x)*(-x^2-2*x-2)^(1/2)/(-(x^2+2*x+2)*(-1+x)^2)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-x^{4} + x^{2} + 2 \, x - 2\right )}^{\frac {3}{2}} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(-x^4+x^2+2*x-2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + x^2 + 2*x - 2)^(3/2)*(x + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x+1\right )\,{\left (-x^4+x^2+2\,x-2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)*(2*x + x^2 - x^4 - 2)^(3/2)),x)

[Out]

int(1/((x + 1)*(2*x + x^2 - x^4 - 2)^(3/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (x - 1\right )^{2} \left (x^{2} + 2 x + 2\right )\right )^{\frac {3}{2}} \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(-x**4+x**2+2*x-2)**(3/2),x)

[Out]

Integral(1/((-(x - 1)**2*(x**2 + 2*x + 2))**(3/2)*(x + 1)), x)

________________________________________________________________________________________