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

Optimal. Leaf size=115 \[ \log \left (\sqrt [3]{x^4-x^3-1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-x^3-1}-x}\right )+\frac {3 \left (x^4-x^3-1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^4-x^3-1} x+\left (x^4-x^3-1\right )^{2/3}\right ) \]

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Rubi [F]  time = 0.92, 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 )^{2/3}}{x^3 \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)^(2/3))/(x^3*(-1 + x^4)),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \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)^(2/3))/(x^3*(-1 + x^4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.54, size = 115, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1-x^3+x^4\right )^{2/3}}{2 x^2}+\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)^(2/3))/(x^3*(-1 + x^4)),x]

[Out]

(3*(-1 - x^3 + x^4)^(2/3))/(2*x^2) + 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 = 5.00, size = 150, normalized size = 1.30 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {728574532 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 812477430 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (355231575 \, x^{4} + 41951449 \, x^{3} - 355231575\right )}}{3 \, {\left (447697125 \, x^{4} - 770525981 \, x^{3} - 447697125\right )}}\right ) + x^{2} \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 (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*x^2*arctan(1/3*(728574532*sqrt(3)*(x^4 - x^3 - 1)^(1/3)*x^2 + 812477430*sqrt(3)*(x^4 - x^3 - 1)
^(2/3)*x + sqrt(3)*(355231575*x^4 + 41951449*x^3 - 355231575))/(447697125*x^4 - 770525981*x^3 - 447697125)) +
x^2*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*(x^4 - x^3 - 1)^(2/
3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 7.62, size = 431, normalized size = 3.75

method	result	size
risch	\(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\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}{\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^{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}{\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^{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}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\)	\(431\)
trager	\(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-8064 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}+15120 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+26868 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -21852 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-50976 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+2002 x^{4}+4353 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}+2532 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-1716 x^{3}+8064 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-26868 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-2002}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )+12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {329472 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-617760 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+2760 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x +52236 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-29628 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-7 x^{4}-1821 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}+2532 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+63 x^{3}-329472 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-2760 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+7}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\)	\(459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^4-x^3-1)^(2/3)/x^2+RootOf(_Z^2+_Z+1)*ln((RootOf(_Z^2+_Z+1)^2*x^3-RootOf(_Z^2+_Z+1)*x^4+3*RootOf(_Z^2+_Z
+1)*(x^4-x^3-1)^(2/3)*x-3*RootOf(_Z^2+_Z+1)*(x^4-x^3-1)^(1/3)*x^2+4*RootOf(_Z^2+_Z+1)*x^3-2*x^4+3*x*(x^4-x^3-1
)^(2/3)-3*(x^4-x^3-1)^(1/3)*x^2+4*x^3+RootOf(_Z^2+_Z+1)+2)/(-1+x)/(1+x)/(x^2+1))-ln((RootOf(_Z^2+_Z+1)^2*x^3+R
ootOf(_Z^2+_Z+1)*x^4-3*RootOf(_Z^2+_Z+1)*(x^4-x^3-1)^(2/3)*x+3*RootOf(_Z^2+_Z+1)*(x^4-x^3-1)^(1/3)*x^2-2*RootO
f(_Z^2+_Z+1)*x^3-x^4+x^3-RootOf(_Z^2+_Z+1)+1)/(-1+x)/(1+x)/(x^2+1))*RootOf(_Z^2+_Z+1)-ln((RootOf(_Z^2+_Z+1)^2*
x^3+RootOf(_Z^2+_Z+1)*x^4-3*RootOf(_Z^2+_Z+1)*(x^4-x^3-1)^(2/3)*x+3*RootOf(_Z^2+_Z+1)*(x^4-x^3-1)^(1/3)*x^2-2*
RootOf(_Z^2+_Z+1)*x^3-x^4+x^3-RootOf(_Z^2+_Z+1)+1)/(-1+x)/(1+x)/(x^2+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} + 3\right )}}{{\left (x^{4} - 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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