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

Optimal. Leaf size=87 \[ \frac {3 \sqrt [3]{x^4-1}}{x}+\log \left (\sqrt [3]{x^4-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.75, 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 {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{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 {\left (3 \sqrt [3]{-1+x^4}\right ) \int \frac {\sqrt [3]{1-x^4}}{x^2} \, dx}{\sqrt [3]{1-x^4}}+\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 \sqrt [3]{-1+x^4} \, _2F_1\left (-\frac {1}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \sqrt [3]{1-x^4}}-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 = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.82, size = 87, 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[((-1 + x^4)^(1/3)*(3 + x^4))/(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)] - L
og[x^2 + x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/2

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fricas [A]  time = 4.05, size = 132, normalized size = 1.52 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (-\frac {14106128635054532 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 89654043956484782 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (35416555940707109 \, x^{4} + 2357401720008016 \, x^{3} - 35416555940707109\right )}}{3 \, {\left (51678794422160641 \, x^{4} + 201291873609016 \, x^{3} - 51678794422160641\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^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan(-1/3*(14106128635054532*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 89654043956484782*sqrt(3)*(x^4 -
1)^(2/3)*x - sqrt(3)*(35416555940707109*x^4 + 2357401720008016*x^3 - 35416555940707109))/(51678794422160641*x
^4 + 201291873609016*x^3 - 51678794422160641)) + 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} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\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^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 8.99, size = 593, normalized size = 6.82

method	result	size
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}-6 \ln \left (\frac {1149120 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}-2154600 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-160116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}-4118136 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +2586522 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1524150 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}-340561 x^{4}-255269 \left (x^{4}-1\right )^{\frac {2}{3}} x +686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-118456 x^{3}-1149120 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+160116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+340561}{x^{4}-x^{3}-1}\right ) \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+\ln \left (\frac {4264416 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}-7995780 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-1524150 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -1531614 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-1684266 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+59850 x^{4}+431087 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+91770 x^{3}-4264416 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+1524150 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-59850}{x^{4}-x^{3}-1}\right )-\ln \left (\frac {1149120 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}-2154600 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-160116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}-4118136 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +2586522 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1524150 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}-340561 x^{4}-255269 \left (x^{4}-1\right )^{\frac {2}{3}} x +686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-118456 x^{3}-1149120 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+160116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+340561}{x^{4}-x^{3}-1}\right )\)	\(593\)
risch	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (\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}-x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+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}+\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+2 \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-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 +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (x^{4}-x^{3}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\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}+x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\left (x^{4}-x^{3}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+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}+x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\left (x^{4}-x^{3}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\right ) \left (\left (x^{4}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{4}-1\right )^{\frac {2}{3}}}\)	\(747\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x^4-1)^(1/3)/x-6*ln((1149120*RootOf(36*_Z^2+6*_Z+1)^2*x^4-2154600*RootOf(36*_Z^2+6*_Z+1)^2*x^3-160116*RootO
f(36*_Z^2+6*_Z+1)*x^4-4118136*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(2/3)*x+2586522*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(1
/3)*x^2+1524150*RootOf(36*_Z^2+6*_Z+1)*x^3-340561*x^4-255269*(x^4-1)^(2/3)*x+686356*(x^4-1)^(1/3)*x^2-118456*x
^3-1149120*RootOf(36*_Z^2+6*_Z+1)^2+160116*RootOf(36*_Z^2+6*_Z+1)+340561)/(x^4-x^3-1))*RootOf(36*_Z^2+6*_Z+1)+
ln((4264416*RootOf(36*_Z^2+6*_Z+1)^2*x^4-7995780*RootOf(36*_Z^2+6*_Z+1)^2*x^3-1524150*RootOf(36*_Z^2+6*_Z+1)*x
^4+4118136*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(2/3)*x-1531614*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(1/3)*x^2-1684266*Roo
tOf(36*_Z^2+6*_Z+1)*x^3+59850*x^4+431087*(x^4-1)^(2/3)*x-686356*(x^4-1)^(1/3)*x^2+91770*x^3-4264416*RootOf(36*
_Z^2+6*_Z+1)^2+1524150*RootOf(36*_Z^2+6*_Z+1)-59850)/(x^4-x^3-1))-ln((1149120*RootOf(36*_Z^2+6*_Z+1)^2*x^4-215
4600*RootOf(36*_Z^2+6*_Z+1)^2*x^3-160116*RootOf(36*_Z^2+6*_Z+1)*x^4-4118136*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(2/
3)*x+2586522*RootOf(36*_Z^2+6*_Z+1)*(x^4-1)^(1/3)*x^2+1524150*RootOf(36*_Z^2+6*_Z+1)*x^3-340561*x^4-255269*(x^
4-1)^(2/3)*x+686356*(x^4-1)^(1/3)*x^2-118456*x^3-1149120*RootOf(36*_Z^2+6*_Z+1)^2+160116*RootOf(36*_Z^2+6*_Z+1
)+340561)/(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} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\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^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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