3.7.41 \(\int \frac {(-1+x^6) (1+x^6)}{\sqrt [4]{x-x^4+x^7} (1+3 x^6+x^{12})} \, dx\)

Optimal. Leaf size=50 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^8+2 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt [4]{x^7-x^4+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

________________________________________________________________________________________

Rubi [F]  time = 1.63, 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^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)),x]

[Out]

(4*x*(1 - (2*x^3)/(1 - I*Sqrt[3]))^(1/4)*(1 - (2*x^3)/(1 + I*Sqrt[3]))^(1/4)*AppellF1[1/4, 1/4, 1/4, 5/4, (2*x
^3)/(1 + I*Sqrt[3]), (2*x^3)/(1 - I*Sqrt[3])])/(3*(x - x^4 + x^7)^(1/4)) - (((2*I)/3)*Sqrt[3 - Sqrt[5]]*x^(1/4
)*(1 - x^3 + x^6)^(1/4)*Defer[Subst][Defer[Int][1/((I*Sqrt[3 - Sqrt[5]] - Sqrt[2]*x^4)*(1 - x^4 + x^8)^(1/4)),
x], x, x^(3/4)])/(x - x^4 + x^7)^(1/4) - (((2*I)/3)*Sqrt[3 + Sqrt[5]]*x^(1/4)*(1 - x^3 + x^6)^(1/4)*Defer[Sub
st][Defer[Int][1/((I*Sqrt[3 + Sqrt[5]] - Sqrt[2]*x^4)*(1 - x^4 + x^8)^(1/4)), x], x, x^(3/4)])/(x - x^4 + x^7)
^(1/4) - (((2*I)/3)*Sqrt[3 - Sqrt[5]]*x^(1/4)*(1 - x^3 + x^6)^(1/4)*Defer[Subst][Defer[Int][1/((I*Sqrt[3 - Sqr
t[5]] + Sqrt[2]*x^4)*(1 - x^4 + x^8)^(1/4)), x], x, x^(3/4)])/(x - x^4 + x^7)^(1/4) - (((2*I)/3)*Sqrt[3 + Sqrt
[5]]*x^(1/4)*(1 - x^3 + x^6)^(1/4)*Defer[Subst][Defer[Int][1/((I*Sqrt[3 + Sqrt[5]] + Sqrt[2]*x^4)*(1 - x^4 + x
^8)^(1/4)), x], x, x^(3/4)])/(x - x^4 + x^7)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x} \sqrt [4]{1-x^3+x^6} \left (1+3 x^6+x^{12}\right )} \, dx}{\sqrt [4]{x-x^4+x^7}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \int \frac {-1+x^{12}}{\sqrt [4]{x} \sqrt [4]{1-x^3+x^6} \left (1+3 x^6+x^{12}\right )} \, dx}{\sqrt [4]{x-x^4+x^7}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {-1+x^{16}}{\sqrt [4]{1-x^4+x^8} \left (1+3 x^8+x^{16}\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1-x^4+x^8}}-\frac {2+3 x^8}{\sqrt [4]{1-x^4+x^8} \left (1+3 x^8+x^{16}\right )}\right ) \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^4+x^8}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {2+3 x^8}{\sqrt [4]{1-x^4+x^8} \left (1+3 x^8+x^{16}\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+\frac {2 x^3}{-1-i \sqrt {3}}} \sqrt [4]{1+\frac {2 x^3}{-1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {2 x^4}{-1-i \sqrt {3}}} \sqrt [4]{1+\frac {2 x^4}{-1+i \sqrt {3}}}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {3-\sqrt {5}}{\sqrt [4]{1-x^4+x^8} \left (3-\sqrt {5}+2 x^8\right )}+\frac {3+\sqrt {5}}{\sqrt [4]{1-x^4+x^8} \left (3+\sqrt {5}+2 x^8\right )}\right ) \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {4 x \sqrt [4]{1-\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [4]{1-\frac {2 x^3}{1+i \sqrt {3}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^3}{1+i \sqrt {3}},\frac {2 x^3}{1-i \sqrt {3}}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \left (3-\sqrt {5}\right ) \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^4+x^8} \left (3-\sqrt {5}+2 x^8\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \left (3+\sqrt {5}\right ) \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^4+x^8} \left (3+\sqrt {5}+2 x^8\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {4 x \sqrt [4]{1-\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [4]{1-\frac {2 x^3}{1+i \sqrt {3}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^3}{1+i \sqrt {3}},\frac {2 x^3}{1-i \sqrt {3}}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \left (3-\sqrt {5}\right ) \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {i}{2 \sqrt {3-\sqrt {5}} \left (i \sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}}+\frac {i}{2 \sqrt {3-\sqrt {5}} \left (i \sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}}\right ) \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (4 \left (3+\sqrt {5}\right ) \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {i}{2 \sqrt {3+\sqrt {5}} \left (i \sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}}+\frac {i}{2 \sqrt {3+\sqrt {5}} \left (i \sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}}\right ) \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ &=\frac {4 x \sqrt [4]{1-\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [4]{1-\frac {2 x^3}{1+i \sqrt {3}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^3}{1+i \sqrt {3}},\frac {2 x^3}{1-i \sqrt {3}}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (2 i \sqrt {3-\sqrt {5}} \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (2 i \sqrt {3-\sqrt {5}} \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (2 i \sqrt {3+\sqrt {5}} \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}-\frac {\left (2 i \sqrt {3+\sqrt {5}} \sqrt [4]{x} \sqrt [4]{1-x^3+x^6}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{1-x^4+x^8}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{x-x^4+x^7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 3.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)),x]

[Out]

Integrate[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.21, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{6} \text {RootSum}\left [2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x-x^4+x^7}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)),x]

[Out]

RootSum[2 + 2*#1^4 + #1^8 & , (-Log[x] + Log[(x - x^4 + x^7)^(1/4) - x*#1])/#1 & ]/6

________________________________________________________________________________________

fricas [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^6-1)*(x^6+1)/(x^7-x^4+x)^(1/4)/(x^12+3*x^6+1),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}}{{\left (x^{12} + 3 \, x^{6} + 1\right )} {\left (x^{7} - x^{4} + x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)*(x^6+1)/(x^7-x^4+x)^(1/4)/(x^12+3*x^6+1),x, algorithm="giac")

[Out]

integrate((x^6 + 1)*(x^6 - 1)/((x^12 + 3*x^6 + 1)*(x^7 - x^4 + x)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{6}-1\right ) \left (x^{6}+1\right )}{\left (x^{7}-x^{4}+x \right )^{\frac {1}{4}} \left (x^{12}+3 x^{6}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)*(x^6+1)/(x^7-x^4+x)^(1/4)/(x^12+3*x^6+1),x)

[Out]

int((x^6-1)*(x^6+1)/(x^7-x^4+x)^(1/4)/(x^12+3*x^6+1),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}}{{\left (x^{12} + 3 \, x^{6} + 1\right )} {\left (x^{7} - x^{4} + x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)*(x^6+1)/(x^7-x^4+x)^(1/4)/(x^12+3*x^6+1),x, algorithm="maxima")

[Out]

integrate((x^6 + 1)*(x^6 - 1)/((x^12 + 3*x^6 + 1)*(x^7 - x^4 + x)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^6-1\right )\,\left (x^6+1\right )}{\left (x^{12}+3\,x^6+1\right )\,{\left (x^7-x^4+x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)*(x^6 + 1))/((3*x^6 + x^12 + 1)*(x - x^4 + x^7)^(1/4)),x)

[Out]

int(((x^6 - 1)*(x^6 + 1))/((3*x^6 + x^12 + 1)*(x - x^4 + x^7)^(1/4)), x)

________________________________________________________________________________________

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**6-1)*(x**6+1)/(x**7-x**4+x)**(1/4)/(x**12+3*x**6+1),x)

[Out]

Timed out

________________________________________________________________________________________