3.19.44 \(\int \frac {(4+x^3) (1+2 x^3+x^6+x^8)}{x^4 \sqrt [4]{1+x^3} (-1-2 x^3-x^6+x^8)} \, dx\)

Optimal. Leaf size=125 \[ \frac {4 \left (x^3+1\right )^{3/4}}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+1}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{x^3+1}}{x}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]

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Rubi [F]  time = 1.59, 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 (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + x^3)*(1 + 2*x^3 + x^6 + x^8))/(x^4*(1 + x^3)^(1/4)*(-1 - 2*x^3 - x^6 + x^8)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx &=\int \left (-\frac {4}{x^4 \sqrt [4]{1+x^3}}-\frac {1}{x \sqrt [4]{1+x^3}}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\left (4 \int \frac {1}{x^4 \sqrt [4]{1+x^3}} \, dx\right )-\int \frac {1}{x \sqrt [4]{1+x^3}} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right )+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}\right ) \, dx+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + x^3)*(1 + 2*x^3 + x^6 + x^8))/(x^4*(1 + x^3)^(1/4)*(-1 - 2*x^3 - x^6 + x^8)),x]

[Out]

Integrate[((4 + x^3)*(1 + 2*x^3 + x^6 + x^8))/(x^4*(1 + x^3)^(1/4)*(-1 - 2*x^3 - x^6 + x^8)), x]

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IntegrateAlgebraic [A]  time = 13.97, size = 125, normalized size = 1.00 \begin {gather*} \frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{-x^2+\sqrt {1+x^3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+x^3}}{x}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4 + x^3)*(1 + 2*x^3 + x^6 + x^8))/(x^4*(1 + x^3)^(1/4)*(-1 - 2*x^3 - x^6 + x^8)),x]

[Out]

(4*(1 + x^3)^(3/4))/(3*x^3) - 2*ArcTan[x/(1 + x^3)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*x*(1 + x^3)^(1/4))/(-x^2 +
Sqrt[1 + x^3])] - 2*ArcTanh[(1 + x^3)^(1/4)/x] + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 +
x^3])]

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fricas [B]  time = 46.86, size = 816, normalized size = 6.53 \begin {gather*} -\frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 12 \, x^{3} \arctan \left (\frac {2 \, {\left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} - 1}\right ) - 12 \, x^{3} \log \left (\frac {x^{4} - 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} + 1} x^{2} - 2 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} - x^{3} - 1}\right ) - 16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="fricas")

[Out]

-1/12*(12*sqrt(2)*x^3*arctan((x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + 2*sqrt(2)*(3*x^5 - x^4 - x)*(x^3 + 1)^(3/4)
+ 2*sqrt(2)*(x^7 - 3*x^6 - 3*x^3)*(x^3 + 1)^(1/4) + 4*(x^6 + x^5 + x^2)*sqrt(x^3 + 1) + (16*(x^3 + 1)^(3/4)*x^
5 + 2*sqrt(2)*(3*x^6 - x^5 - x^2)*sqrt(x^3 + 1) + sqrt(2)*(x^8 + 8*x^7 - x^6 + 8*x^4 - 2*x^3 - 1) + 4*(x^7 + x
^6 + x^3)*(x^3 + 1)^(1/4))*sqrt((x^4 - 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 - 2*sqrt(2)*(
x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3 + 1)) + 1)/(x^8 - 14*x^7 + x^6 - 14*x^4 + 2*x^3 + 1)) - 12*sqrt(2)*x^3*arctan
((x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 - 2*sqrt(2)*(3*x^5 - x^4 - x)*(x^3 + 1)^(3/4) - 2*sqrt(2)*(x^7 - 3*x^6 - 3
*x^3)*(x^3 + 1)^(1/4) + 4*(x^6 + x^5 + x^2)*sqrt(x^3 + 1) + (16*(x^3 + 1)^(3/4)*x^5 - 2*sqrt(2)*(3*x^6 - x^5 -
x^2)*sqrt(x^3 + 1) - sqrt(2)*(x^8 + 8*x^7 - x^6 + 8*x^4 - 2*x^3 - 1) + 4*(x^7 + x^6 + x^3)*(x^3 + 1)^(1/4))*s
qrt((x^4 + 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 + 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 +
x^3 + 1)) + 1)/(x^8 - 14*x^7 + x^6 - 14*x^4 + 2*x^3 + 1)) - 3*sqrt(2)*x^3*log(4*(x^4 + 2*sqrt(2)*(x^3 + 1)^(1
/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 + 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3 + 1)) + 3*sqrt(2)*x^3*log(4*
(x^4 - 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 - 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3
+ 1)) - 12*x^3*arctan(2*((x^3 + 1)^(1/4)*x^3 + (x^3 + 1)^(3/4)*x)/(x^4 - x^3 - 1)) - 12*x^3*log((x^4 - 2*(x^3
+ 1)^(1/4)*x^3 + x^3 + 2*sqrt(x^3 + 1)*x^2 - 2*(x^3 + 1)^(3/4)*x + 1)/(x^4 - x^3 - 1)) - 16*(x^3 + 1)^(3/4))/
x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 4)/((x^8 - x^6 - 2*x^3 - 1)*(x^3 + 1)^(1/4)*x^4), x)

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maple [C]  time = 18.01, size = 435, normalized size = 3.48

method	result	size
risch	\(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x +2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}-1}\right )-\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+x^{3}+1}{x^{4}-x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )\)	\(435\)
trager	\(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {3}{4}} x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-2 x^{2} \sqrt {x^{3}+1}+x^{4}+x^{3}+1}{x^{4}-x^{3}-1}\right )-\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+x^{3}+1}{x^{4}-x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )\)	\(439\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

4/3*(x^3+1)^(3/4)/x^3-RootOf(_Z^2+1)*ln((2*RootOf(_Z^2+1)*(x^3+1)^(1/2)*x^2-RootOf(_Z^2+1)*x^4-RootOf(_Z^2+1)*
x^3-2*(x^3+1)^(3/4)*x+2*(x^3+1)^(1/4)*x^3-RootOf(_Z^2+1))/(x^4-x^3-1))-ln(-(2*(x^3+1)^(3/4)*x+2*x^2*(x^3+1)^(1
/2)+2*(x^3+1)^(1/4)*x^3+x^4+x^3+1)/(x^4-x^3-1))+RootOf(_Z^2+RootOf(_Z^2+1))*ln((2*RootOf(_Z^2+1)*RootOf(_Z^2+R
ootOf(_Z^2+1))*(x^3+1)^(1/2)*x^2+2*RootOf(_Z^2+1)*(x^3+1)^(1/4)*x^3-RootOf(_Z^2+RootOf(_Z^2+1))*x^4+2*(x^3+1)^
(3/4)*x+RootOf(_Z^2+RootOf(_Z^2+1))*x^3+RootOf(_Z^2+RootOf(_Z^2+1)))/(x^4+x^3+1))-RootOf(_Z^2+1)*RootOf(_Z^2+R
ootOf(_Z^2+1))*ln(-(-RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^4+2*RootOf(_Z^2+1)*(x^3+1)^(1/4)*x^3+RootOf(
_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+2*RootOf(_Z^2+RootOf(_Z^2+1))*(x^3+1)^(1/2)*x^2-2*(x^3+1)^(3/4)*x+Root
Of(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+1)))/(x^4+x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 4)/((x^8 - x^6 - 2*x^3 - 1)*(x^3 + 1)^(1/4)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^3+4\right )\,\left (x^8+x^6+2\,x^3+1\right )}{x^4\,{\left (x^3+1\right )}^{1/4}\,\left (-x^8+x^6+2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 4)*(2*x^3 + x^6 + x^8 + 1))/(x^4*(x^3 + 1)^(1/4)*(2*x^3 + x^6 - x^8 + 1)),x)

[Out]

int(-((x^3 + 4)*(2*x^3 + x^6 + x^8 + 1))/(x^4*(x^3 + 1)^(1/4)*(2*x^3 + x^6 - x^8 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+4)*(x**8+x**6+2*x**3+1)/x**4/(x**3+1)**(1/4)/(x**8-x**6-2*x**3-1),x)

[Out]

Timed out

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