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

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 = 0.83, size = 87, normalized size = 1.00 \begin {gather*} \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 ) \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 = 3.82, 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 = 3.03, size = 754, normalized size = 8.67

result too large to display

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)

[Out]

3*(x^4-1)^(1/3)/x+(RootOf(_Z^2+_Z+1)*ln(-(RootOf(_Z^2+_Z+1)^2*x^7-RootOf(_Z^2+_Z+1)*x^8+x^7*RootOf(_Z^2+_Z+1)-
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-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-RootOf(_Z^2+_Z+1)*x^3-2*(x^8-2*x^4+1)^(2/3)*x^2+2*x^4+Root
Of(_Z^2+_Z+1)*(x^8-2*x^4+1)^(1/3)*x+2*(x^8-2*x^4+1)^(1/3)*x-RootOf(_Z^2+_Z+1)-1)/(x^4-x^3-1)/(-1+x)/(1+x)/(x^2
+1))-ln(-(RootOf(_Z^2+_Z+1)^2*x^7+RootOf(_Z^2+_Z+1)*x^8+x^7*RootOf(_Z^2+_Z+1)+RootOf(_Z^2+_Z+1)*(x^8-2*x^4+1)^
(1/3)*x^5-(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-RootOf(_Z^2+_Z+1)*x^3-(x^8-2*x^4+1)^(2/3)*x^2-RootOf(_Z^2+_Z+1)*(x^8-2*x^4+1)^(1/3)*x+(x^8-2*x^4
+1)^(1/3)*x+RootOf(_Z^2+_Z+1))/(x^4-x^3-1)/(-1+x)/(1+x)/(x^2+1))*RootOf(_Z^2+_Z+1)-ln(-(RootOf(_Z^2+_Z+1)^2*x^
7+RootOf(_Z^2+_Z+1)*x^8+x^7*RootOf(_Z^2+_Z+1)+RootOf(_Z^2+_Z+1)*(x^8-2*x^4+1)^(1/3)*x^5-(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-RootOf(_Z^2+_Z+1)*
x^3-(x^8-2*x^4+1)^(2/3)*x^2-RootOf(_Z^2+_Z+1)*(x^8-2*x^4+1)^(1/3)*x+(x^8-2*x^4+1)^(1/3)*x+RootOf(_Z^2+_Z+1))/(
x^4-x^3-1)/(-1+x)/(1+x)/(x^2+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} + 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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