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

Optimal. Leaf size=90 \[ \frac {3 \sqrt [3]{x^4+1}}{x}-\log \left (\sqrt [3]{x^4+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.74, 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 ) \sqrt [3]{1+x^4}}{x^2 \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)^(1/3))/(x^2*(1 + x^3 + x^4)),x]

[Out]

(3*Hypergeometric2F1[-1/3, -1/4, 3/4, -x^4])/x + 3*Defer[Int][(x*(1 + x^4)^(1/3))/(1 + x^3 + x^4), x] + 4*Defe
r[Int][(x^2*(1 + x^4)^(1/3))/(1 + x^3 + x^4), x]

Rubi steps

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.87, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{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)^(1/3))/(x^2*(1 + x^3 + x^4)),x]

[Out]

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

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fricas [A]  time = 2.86, size = 122, normalized size = 1.36 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}}{3 \, {\left (x^{4} - x^{3} + 1\right )}}\right ) - x \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 ) + 6 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*x*arctan(1/3*(2*sqrt(3)*(x^4 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^4 + 1)^(2/3)*x + sqrt(3)*(x^4 + x^3
+ 1))/(x^4 - x^3 + 1)) - x*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))
+ 6*(x^4 + 1)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.08, size = 477, normalized size = 5.30

method	result	size
risch	\(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -1}{\left (x^{4}+x^{3}+1\right ) \left (x^{4}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{7}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+x^{7}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+2 \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +x^{3}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (x^{4}+x^{3}+1\right ) \left (x^{4}+1\right )}\right )\right ) \left (\left (x^{4}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{4}+1\right )^{\frac {2}{3}}}\)	\(477\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}+42 \left (x^{4}+1\right )^{\frac {2}{3}} x -42 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-39 \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 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+111 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-14 x^{4}+27 \left (x^{4}+1\right )^{\frac {2}{3}} x -27 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-14 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-14}{x^{4}+x^{3}+1}\right )+\ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}+42 \left (x^{4}+1\right )^{\frac {2}{3}} x -42 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-10}{x^{4}+x^{3}+1}\right )\)	\(598\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x^4+1)^(1/3)/x+(-ln(-(RootOf(_Z^2-_Z+1)*x^7-x^8+RootOf(_Z^2-_Z+1)*(x^8+2*x^4+1)^(1/3)*x^5-2*(x^8+2*x^4+1)^(
1/3)*x^5+2*RootOf(_Z^2-_Z+1)*(x^8+2*x^4+1)^(2/3)*x^2+RootOf(_Z^2-_Z+1)*x^3-(x^8+2*x^4+1)^(2/3)*x^2-2*x^4+RootO
f(_Z^2-_Z+1)*(x^8+2*x^4+1)^(1/3)*x-2*(x^8+2*x^4+1)^(1/3)*x-1)/(x^4+x^3+1)/(x^4+1))+RootOf(_Z^2-_Z+1)*ln((RootO
f(_Z^2-_Z+1)^2*x^7+RootOf(_Z^2-_Z+1)*x^8-2*RootOf(_Z^2-_Z+1)*x^7-x^8+RootOf(_Z^2-_Z+1)*(x^8+2*x^4+1)^(1/3)*x^5
+x^7+(x^8+2*x^4+1)^(1/3)*x^5+RootOf(_Z^2-_Z+1)^2*x^3-RootOf(_Z^2-_Z+1)*(x^8+2*x^4+1)^(2/3)*x^2+2*RootOf(_Z^2-_
Z+1)*x^4-2*RootOf(_Z^2-_Z+1)*x^3+2*(x^8+2*x^4+1)^(2/3)*x^2-2*x^4+RootOf(_Z^2-_Z+1)*(x^8+2*x^4+1)^(1/3)*x+x^3+(
x^8+2*x^4+1)^(1/3)*x+RootOf(_Z^2-_Z+1)-1)/(x^4+x^3+1)/(x^4+1)))/(x^4+1)^(2/3)*((x^4+1)^2)^(1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

Timed out

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