3.15.75 \(\int \frac {(-3+x^4) (1+x^4)^{2/3} (1+2 x^3+x^4)}{x^6 (1-x^3+x^4)} \, dx\)

Optimal. Leaf size=104 \[ 3 \log \left (\sqrt [3]{x^4+1}-x\right )-3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}+x}\right )-\frac {3}{2} \log \left (\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right )+\frac {3 \left (x^4+1\right )^{2/3} \left (2 x^4+15 x^3+2\right )}{10 x^5} \]

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Rubi [F]  time = 1.41, 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 (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*(1 + x^4)^(2/3))/(2*x^2) + (18*x^2)/(1 - Sqrt[3] - (1 + x^4)^(1/3)) - (9*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1
+ x^4)^(1/3))*Sqrt[(1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2]*EllipticE[ArcSin
[(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))], -7 + 4*Sqrt[3]])/(x^2*Sqrt[-((1 - (1 + x^4)
^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2)]) + (6*Sqrt[2]*3^(3/4)*(1 - (1 + x^4)^(1/3))*Sqrt[(1 + (1 + x^4)^(1
/3) + (1 + x^4)^(2/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1
- Sqrt[3] - (1 + x^4)^(1/3))], -7 + 4*Sqrt[3]])/(x^2*Sqrt[-((1 - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/
3))^2)]) + (3*Hypergeometric2F1[-5/4, -2/3, -1/4, -x^4])/(5*x^5) - Hypergeometric2F1[-2/3, -1/4, 3/4, -x^4]/x
- 9*Defer[Int][(1 + x^4)^(2/3)/(1 - x^3 + x^4), x] + 12*Defer[Int][(x*(1 + x^4)^(2/3))/(1 - x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{x^6}-\frac {9 \left (1+x^4\right )^{2/3}}{x^3}+\frac {\left (1+x^4\right )^{2/3}}{x^2}+\frac {3 (-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^4\right )^{2/3}}{x^6} \, dx\right )+3 \int \frac {(-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-9 \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^4\right )^{2/3}}{x^2} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )}{5 x^5}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}+3 \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{1-x^3+x^4}+\frac {4 x \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx-\frac {9}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=\frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )}{5 x^5}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}-6 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx\\ &=\frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )}{5 x^5}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-\frac {\left (9 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}\\ &=\frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )}{5 x^5}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+\frac {\left (9 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}-\frac {\left (9 \sqrt {2 \left (2+\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}\\ &=\frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {18 x^2}{1-\sqrt {3}-\sqrt [3]{1+x^4}}-\frac {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {6 \sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )}{5 x^5}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.50, size = 104, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^4\right )^{2/3} \left (2+15 x^3+2 x^4\right )}{10 x^5}-3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+3 \log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {3}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + x^4)^(2/3)*(2 + 15*x^3 + 2*x^4))/(10*x^5) - 3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))] + 3*
Log[-x + (1 + x^4)^(1/3)] - (3*Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/2

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fricas [A]  time = 3.19, size = 146, normalized size = 1.40 \begin {gather*} -\frac {3 \, {\left (10 \, \sqrt {3} x^{5} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) - {\left (2 \, x^{4} + 15 \, x^{3} + 2\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}\right )}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/10*(10*sqrt(3)*x^5*arctan(-1/3*(13034*sqrt(3)*(x^4 + 1)^(1/3)*x^2 - 686*sqrt(3)*(x^4 + 1)^(2/3)*x + sqrt(3)
*(37*x^4 + 6137*x^3 + 37))/(x^4 + 6859*x^3 + 1)) - 5*x^5*log((x^4 - x^3 + 3*(x^4 + 1)^(1/3)*x^2 - 3*(x^4 + 1)^
(2/3)*x + 1)/(x^4 - x^3 + 1)) - (2*x^4 + 15*x^3 + 2)*(x^4 + 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 6.92, size = 292, normalized size = 2.81

method	result	size
risch	\(\frac {\frac {3}{5} x^{8}+\frac {6}{5} x^{4}+\frac {3}{5}+\frac {9}{2} x^{7}+\frac {9}{2} x^{3}}{x^{5} \left (x^{4}+1\right )^{\frac {1}{3}}}+3 \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}+1}\right )+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +2 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}+1}\right )\)	\(292\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (2 x^{4}+15 x^{3}+2\right )}{10 x^{5}}-9 \ln \left (-\frac {72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-144 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +60 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-102 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}-20 \left (x^{4}+1\right )^{\frac {2}{3}} x +37 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-15 x^{3}+72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10}{x^{4}-x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \ln \left (\frac {12357 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-24714 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+8922 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -33651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+10989 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3435 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x -8010 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1145 x^{3}+12357 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+8922 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3435}{x^{4}-x^{3}+1}\right )-3 \ln \left (-\frac {72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-144 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +60 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-102 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}-20 \left (x^{4}+1\right )^{\frac {2}{3}} x +37 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-15 x^{3}+72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10}{x^{4}-x^{3}+1}\right )\)	\(609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*x^8+15*x^7+4*x^4+15*x^3+2)/x^5/(x^4+1)^(1/3)+3*ln((RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*x^4-RootO
f(_Z^2+_Z+1)*(x^4+1)^(2/3)*x-(x^4+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+2*RootOf(_Z^2+_Z+1)*x^3+x^4-2*(x^4+1)^(2/3)*x
+x^2*(x^4+1)^(1/3)+x^3+RootOf(_Z^2+_Z+1)+1)/(x^4-x^3+1))+3*RootOf(_Z^2+_Z+1)*ln(-(RootOf(_Z^2+_Z+1)^2*x^3-Root
Of(_Z^2+_Z+1)*x^4-RootOf(_Z^2+_Z+1)*(x^4+1)^(2/3)*x+2*(x^4+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^
3-x^4+(x^4+1)^(2/3)*x+x^2*(x^4+1)^(1/3)-RootOf(_Z^2+_Z+1)-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^{4} + 2 \, x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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