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

Optimal. Leaf size=100 \[ 2 \log \left (\sqrt [3]{x^4-1}-x\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-1}+x}\right )-\log \left (\sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+x^2\right )+\frac {3 \left (x^4-1\right )^{2/3} \left (x^4+5 x^3-1\right )}{5 x^5} \]

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Rubi [F]  time = 1.35, 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 (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+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[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-1 - x^3 + x^4)),x]

[Out]

(3*(-1 + x^4)^(2/3))/x^2 - (12*x^2)/(1 + Sqrt[3] + (-1 + x^4)^(1/3)) + (6*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[ArcS
in[(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]) - (4*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*(-1 + x^4)^(2/3)*Hypergeometric2F1[-5/4, -2/3, -1/4, x^4])/(5*x^5*(1 - x^4)^(2/3)) - ((-
1 + x^4)^(2/3)*Hypergeometric2F1[-2/3, -1/4, 3/4, x^4])/(x*(1 - x^4)^(2/3)) - 6*Defer[Int][(-1 + x^4)^(2/3)/(-
1 - x^3 + x^4), x] + 8*Defer[Int][(x*(-1 + x^4)^(2/3))/(-1 - x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+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 {6 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {\left (-1+x^4\right )^{2/3}}{x^2}+\frac {2 (-3+4 x) \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4}\right ) \, dx\\ &=2 \int \frac {(-3+4 x) \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx-6 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx\\ &=2 \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-3 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{\left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{\left (1-x^4\right )^{2/3}}\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx-\frac {\left (6 \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 {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx-\frac {\left (6 \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 (6 \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 {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {12 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}+\frac {6 \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 {4 \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 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \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 = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+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[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-1 - x^3 + x^4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 3.39, size = 100, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^4\right )^{2/3} \left (-1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{-1+x^4}\right )-\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)^(2/3)*(3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-1 - x^3 + x^4)),x]

[Out]

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

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fricas [A]  time = 5.00, size = 147, normalized size = 1.47 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \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 ) - 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 ) - 3 \, {\left (x^{4} + 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(10*sqrt(3)*x^5*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))/(516787944221606
41*x^4 + 201291873609016*x^3 - 51678794422160641)) - 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)) - 3*(x^4 + 5*x^3 - 1)*(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} + x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\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^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 8.56, size = 321, normalized size = 3.21

method	result	size
risch	\(\frac {\frac {3}{5} x^{8}-\frac {6}{5} x^{4}+\frac {3}{5}+3 x^{7}-3 x^{3}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x +2 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+x^{3}-1}{x^{4}-x^{3}-1}\right )-2 \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x -\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1}{x^{4}-x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x -\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+1}{x^{4}-x^{3}-1}\right )\)	\(321\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (x^{4}+5 x^{3}-1\right )}{5 x^{5}}+2 \ln \left (\frac {1091690496 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{4}-2046919680 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}-24386400 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}[phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -24505824 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-26948256 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+59850 x^{4}+431087 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+91770 x^{3}-1091690496 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-59850}{x^{4}-x^{3}-1}\right )+192 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (\frac {294174720 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{4}-551577600 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}-2561856 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}-65890176 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-340561 x^{4}-255269 \left (x^{4}-1\right )^{\frac {2}{3}} x +686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-118456 x^{3}-294174720 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+2561856 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+340561}{x^{4}-x^{3}-1}\right )\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^8+5*x^7-2*x^4-5*x^3+1)/x^5/(x^4-1)^(1/3)+2*RootOf(_Z^2+_Z+1)*ln((RootOf(_Z^2+_Z+1)*(x^4-1)^(2/3)*x+(x^4
-1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^3+x^4+2*(x^4-1)^(2/3)*x+2*(x^4-1)^(1/3)*x^2+x^3-1)/(x^4-x^
3-1))-2*ln(-(RootOf(_Z^2+_Z+1)*(x^4-1)^(2/3)*x+(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^4-1)^(1/3)*x^2+1)/(x^4-x^3-1))*RootOf(_Z^2+_Z+1)-2*ln(-(RootOf(_Z^2+_Z+1)*(x^4-1)^(2/3)*x+(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^4-1)^(1/3)*x^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^{4} + x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\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^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Timed out

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