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

Optimal. Leaf size=118 \[ -\log \left (\sqrt [3]{x^4+x^3+1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^3+1}+x}\right )+\frac {3 \left (x^4+x^3+1\right )^{2/3} \left (2 x^4-3 x^3+2\right )}{10 x^5}+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+x^3+1} x+\left (x^4+x^3+1\right )^{2/3}\right ) \]

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Rubi [F]  time = 1.63, 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^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.08, size = 118, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^3+x^4\right )^{2/3} \left (2-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{1+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^4}+\left (1+x^3+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 5.58, size = 153, normalized size = 1.30 \begin {gather*} \frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {7043582 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 984256 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (145408 \, x^{4} + 3029663 \, x^{3} + 145408\right )}}{32768 \, x^{4} + 12041757 \, x^{3} + 32768}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} + 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + 1}\right ) + 3 \, {\left (2 \, x^{4} - 3 \, x^{3} + 2\right )} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(10*sqrt(3)*x^5*arctan(-(7043582*sqrt(3)*(x^4 + x^3 + 1)^(1/3)*x^2 - 984256*sqrt(3)*(x^4 + x^3 + 1)^(2/3)
*x + sqrt(3)*(145408*x^4 + 3029663*x^3 + 145408))/(32768*x^4 + 12041757*x^3 + 32768)) - 5*x^5*log((x^4 + 3*(x^
4 + x^3 + 1)^(1/3)*x^2 - 3*(x^4 + x^3 + 1)^(2/3)*x + 1)/(x^4 + 1)) + 3*(2*x^4 - 3*x^3 + 2)*(x^4 + x^3 + 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} + x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{4} + 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-x^3+1)*(x^4+x^3+1)^(2/3)/x^6/(x^4+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 6.02, size = 435, normalized size = 3.69

method	result	size
trager	\(\frac {3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \left (2 x^{4}-3 x^{3}+2\right )}{10 x^{5}}-\ln \left (\frac {31401 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-62802 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9177 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+37851 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -134634 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+117717 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-22224 x^{4}+32261 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}+12617 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-51856 x^{3}+31401 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+9177 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-22224}{x^{4}+1}\right )+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-133344 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-99363 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+37851 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +96783 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-90186 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+13956 x^{4}-44878 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}+12617 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+17445 x^{3}+66672 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-99363 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+13956}{x^{4}+1}\right )\)	\(435\)
risch	\(\frac {\frac {3}{5} x^{8}-\frac {3}{10} x^{7}-\frac {9}{10} x^{6}+\frac {6}{5} x^{4}-\frac {3}{10} x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 x^{4}-3 x \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}-3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2}{x^{4}+1}\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\right )\)	\(442\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(x^4+x^3+1)^(2/3)*(2*x^4-3*x^3+2)/x^5-ln((31401*RootOf(9*_Z^2-3*_Z+1)^2*x^4-62802*RootOf(9*_Z^2-3*_Z+1)^2
*x^3+9177*RootOf(9*_Z^2-3*_Z+1)*x^4+37851*RootOf(9*_Z^2-3*_Z+1)*(x^4+x^3+1)^(2/3)*x-134634*(x^4+x^3+1)^(1/3)*R
ootOf(9*_Z^2-3*_Z+1)*x^2+117717*RootOf(9*_Z^2-3*_Z+1)*x^3-22224*x^4+32261*x*(x^4+x^3+1)^(2/3)+12617*(x^4+x^3+1
)^(1/3)*x^2-51856*x^3+31401*RootOf(9*_Z^2-3*_Z+1)^2+9177*RootOf(9*_Z^2-3*_Z+1)-22224)/(x^4+1))+3*RootOf(9*_Z^2
-3*_Z+1)*ln(-(66672*RootOf(9*_Z^2-3*_Z+1)^2*x^4-133344*RootOf(9*_Z^2-3*_Z+1)^2*x^3-99363*RootOf(9*_Z^2-3*_Z+1)
*x^4+37851*RootOf(9*_Z^2-3*_Z+1)*(x^4+x^3+1)^(2/3)*x+96783*(x^4+x^3+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2-90186*R
ootOf(9*_Z^2-3*_Z+1)*x^3+13956*x^4-44878*x*(x^4+x^3+1)^(2/3)+12617*(x^4+x^3+1)^(1/3)*x^2+17445*x^3+66672*RootO
f(9*_Z^2-3*_Z+1)^2-99363*RootOf(9*_Z^2-3*_Z+1)+13956)/(x^4+1))

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

[Out]

integrate((x^4 + x^3 + 1)^(2/3)*(x^4 - x^3 + 1)*(x^4 - 3)/((x^4 + 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^4-3\right )\,{\left (x^4+x^3+1\right )}^{2/3}\,\left (x^4-x^3+1\right )}{x^6\,\left (x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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