3.9.99 \(\int \frac {(2+x^2) \sqrt [4]{-1-x^2+x^4} (1+x^2+x^4)}{x^6 (1+x^2)} \, dx\)

Optimal. Leaf size=68 \[ -\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2-1}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2-1}}\right )-\frac {2 \sqrt [4]{x^4-x^2-1} \left (4 x^4+x^2+1\right )}{5 x^5} \]

________________________________________________________________________________________

Rubi [F]  time = 6.84, 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 {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2 + x^2)*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + x^4))/(x^6*(1 + x^2)),x]

[Out]

(-2*(-1 - x^2 + x^4)^(1/4)*AppellF1[-1/2, -1/4, -1/4, 1/2, (2*x^2)/(1 + Sqrt[5]), (2*x^2)/(1 - Sqrt[5])])/(x*(
1 - (2*x^2)/(1 - Sqrt[5]))^(1/4)*(1 - (2*x^2)/(1 + Sqrt[5]))^(1/4)) - ((1 - (2*x^2)/(1 - Sqrt[5]))^(5/4)*(-1 -
x^2 + x^4)^(1/4)*Hypergeometric2F1[-3/2, -1/4, -1/2, (-2*(x^2/(1 - Sqrt[5]) - x^2/(1 + Sqrt[5])))/(1 - (2*x^2
)/(1 - Sqrt[5]))])/(3*x^3*(1 - (2*x^2)/(1 + Sqrt[5]))^(1/4)) + (4*(1 - (2*x^2)/(1 - Sqrt[5]))*(-1 - x^2 + x^4)
^(1/4)*((3*(1 - Sqrt[5]) + (13 - 3*Sqrt[5])*x^2 + 2*(1 - Sqrt[5])*x^4)*Gamma[-1/4]*Hypergeometric2F1[-1/4, 1,
-1/2, (-2*Sqrt[5]*x^2)/(2 + (1 - Sqrt[5])*x^2)] + 4*x^2*(5 - Sqrt[5] + 2*Sqrt[5]*x^2)*Gamma[3/4]*Hypergeometri
c2F1[3/4, 2, 1/2, (-2*Sqrt[5]*x^2)/(2 + (1 - Sqrt[5])*x^2)]))/(15*(3 - Sqrt[5])*x^5*(1 + Sqrt[5] - 2*x^2)*Gamm
a[-1/4]) + Defer[Int][(-1 - x^2 + x^4)^(1/4)/(-1 - x^2), x]

Rubi steps

\begin {align*} \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx &=\int \left (\frac {2 \sqrt [4]{-1-x^2+x^4}}{x^6}+\frac {\sqrt [4]{-1-x^2+x^4}}{x^4}+\frac {2 \sqrt [4]{-1-x^2+x^4}}{x^2}+\frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt [4]{-1-x^2+x^4}}{x^6} \, dx+2 \int \frac {\sqrt [4]{-1-x^2+x^4}}{x^2} \, dx+\int \frac {\sqrt [4]{-1-x^2+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ &=\frac {\sqrt [4]{-1-x^2+x^4} \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^4} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\frac {\left (2 \sqrt [4]{-1-x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^6} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\frac {\left (2 \sqrt [4]{-1-x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ &=-\frac {2 \sqrt [4]{-1-x^2+x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};\frac {2 x^2}{1+\sqrt {5}},\frac {2 x^2}{1-\sqrt {5}}\right )}{x \sqrt [4]{1-\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (1-\frac {2 x^2}{1-\sqrt {5}}\right )^{5/4} \sqrt [4]{-1-x^2+x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};-\frac {1}{2};-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{1-\frac {2 x^2}{1-\sqrt {5}}}\right )}{3 x^3 \sqrt [4]{1-\frac {2 x^2}{1+\sqrt {5}}}}+\frac {4 \left (1-\frac {2 x^2}{1-\sqrt {5}}\right ) \sqrt [4]{-1-x^2+x^4} \left (\left (3 \left (1-\sqrt {5}\right )+\left (13-3 \sqrt {5}\right ) x^2+2 \left (1-\sqrt {5}\right ) x^4\right ) \Gamma \left (-\frac {1}{4}\right ) \, _2F_1\left (-\frac {1}{4},1;-\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2+\left (1-\sqrt {5}\right ) x^2}\right )+4 x^2 \left (5-\sqrt {5}+2 \sqrt {5} x^2\right ) \Gamma \left (\frac {3}{4}\right ) \, _2F_1\left (\frac {3}{4},2;\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2+\left (1-\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3-\sqrt {5}\right ) x^5 \left (1+\sqrt {5}-2 x^2\right ) \Gamma \left (-\frac {1}{4}\right )}+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 4.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((2 + x^2)*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + x^4))/(x^6*(1 + x^2)),x]

[Out]

Integrate[((2 + x^2)*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + x^4))/(x^6*(1 + x^2)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.28, size = 68, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{-1-x^2+x^4} \left (1+x^2+4 x^4\right )}{5 x^5}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + x^2)*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + x^4))/(x^6*(1 + x^2)),x]

[Out]

(-2*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + 4*x^4))/(5*x^5) - ArcTan[x/(-1 - x^2 + x^4)^(1/4)] + ArcTanh[x/(-1 - x^2
+ x^4)^(1/4)]

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)*(x^4-x^2-1)^(1/4)*(x^4+x^2+1)/x^6/(x^2+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)*(x^4-x^2-1)^(1/4)*(x^4+x^2+1)/x^6/(x^2+1),x, algorithm="giac")

[Out]

integrate((x^4 + x^2 + 1)*(x^4 - x^2 - 1)^(1/4)*(x^2 + 2)/((x^2 + 1)*x^6), x)

________________________________________________________________________________________

maple [C]  time = 8.74, size = 201, normalized size = 2.96

method	result	size
trager	\(-\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} \left (4 x^{4}+x^{2}+1\right )}{5 x^{5}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-x^{2}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x -2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{2}+\frac {\ln \left (\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x +2 \sqrt {x^{4}-x^{2}-1}\, x^{2}+2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 x^{4}-x^{2}-1}{x^{2}+1}\right )}{2}\)	\(201\)
risch	\(-\frac {2 \left (4 x^{8}-3 x^{6}-4 x^{4}-2 x^{2}-1\right )}{5 x^{5} \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {\ln \left (\frac {-2 x^{12}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+5 x^{10}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}-4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+x^{8}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-7 x^{6}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}+4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-x^{4}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x +3 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 x^{12}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+5 x^{10}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}+x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-7 x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}-x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x +3 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}\right ) \left (\left (x^{4}-x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}\)	\(700\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2)*(x^4-x^2-1)^(1/4)*(x^4+x^2+1)/x^6/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

-2/5*(x^4-x^2-1)^(1/4)*(4*x^4+x^2+1)/x^5+1/2*RootOf(_Z^2+1)*ln((2*RootOf(_Z^2+1)*(x^4-x^2-1)^(1/2)*x^2-2*RootO
f(_Z^2+1)*x^4+2*(x^4-x^2-1)^(3/4)*x-2*(x^4-x^2-1)^(1/4)*x^3+RootOf(_Z^2+1)*x^2+RootOf(_Z^2+1))/(x^2+1))+1/2*ln
((2*(x^4-x^2-1)^(3/4)*x+2*(x^4-x^2-1)^(1/2)*x^2+2*(x^4-x^2-1)^(1/4)*x^3+2*x^4-x^2-1)/(x^2+1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)*(x^4-x^2-1)^(1/4)*(x^4+x^2+1)/x^6/(x^2+1),x, algorithm="maxima")

[Out]

integrate((x^4 + x^2 + 1)*(x^4 - x^2 - 1)^(1/4)*(x^2 + 2)/((x^2 + 1)*x^6), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2+2\right )\,\left (x^4+x^2+1\right )\,{\left (x^4-x^2-1\right )}^{1/4}}{x^6\,\left (x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 + 2)*(x^2 + x^4 + 1)*(x^4 - x^2 - 1)^(1/4))/(x^6*(x^2 + 1)),x)

[Out]

int(((x^2 + 2)*(x^2 + x^4 + 1)*(x^4 - x^2 - 1)^(1/4))/(x^6*(x^2 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 2\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \sqrt [4]{x^{4} - x^{2} - 1}}{x^{6} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2)*(x**4-x**2-1)**(1/4)*(x**4+x**2+1)/x**6/(x**2+1),x)

[Out]

Integral((x**2 + 2)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 - 1)**(1/4)/(x**6*(x**2 + 1)), x)

________________________________________________________________________________________