3.29.8 \(\int \frac {(4+x^3) (1+x^3+x^4)}{\sqrt [4]{1+x^3} (1+2 x^3+x^6+x^8)} \, dx\)

Optimal. Leaf size=275 \[ -\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt {2-\sqrt {2}} x^2}{\sqrt {2}-2}-\frac {\sqrt {2-\sqrt {2}} \sqrt {x^3+1}}{\sqrt {2}-2}}{x \sqrt [4]{x^3+1}}\right )-\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt {x^3+1}}{\sqrt {2+\sqrt {2}}}-\frac {x^2}{\sqrt {2+\sqrt {2}}}}{x \sqrt [4]{x^3+1}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.23, 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 (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + x^3)*(1 + x^3 + x^4))/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)),x]

[Out]

4*Defer[Int][1/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)), x] + 5*Defer[Int][x^3/((1 + x^3)^(1/4)*(1 + 2*x^3 +
x^6 + x^8)), x] + 4*Defer[Int][x^4/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)), x] + Defer[Int][x^6/((1 + x^3)^(
1/4)*(1 + 2*x^3 + x^6 + x^8)), x] + Defer[Int][x^7/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)), x]

Rubi steps

\begin {align*} \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx &=\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {5 x^3}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {4 x^4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {x^6}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}+\frac {x^7}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+4 \int \frac {x^4}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+5 \int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+\int \frac {x^6}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx+\int \frac {x^7}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + x^3)*(1 + x^3 + x^4))/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)),x]

[Out]

Integrate[((4 + x^3)*(1 + x^3 + x^4))/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.33, size = 257, normalized size = 0.93 \begin {gather*} -\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {1-\frac {1}{\sqrt {2}}} x^2+\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {1+x^3}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {1+\frac {1}{\sqrt {2}}} x^2+\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {1+x^3}}{x \sqrt [4]{1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4 + x^3)*(1 + x^3 + x^4))/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)),x]

[Out]

-(Sqrt[(2 - Sqrt[2])/2]*ArcTan[(-(Sqrt[1 - 1/Sqrt[2]]*x^2) + Sqrt[1 - 1/Sqrt[2]]*Sqrt[1 + x^3])/(x*(1 + x^3)^(
1/4))]) - Sqrt[(2 + Sqrt[2])/2]*ArcTan[(-(Sqrt[1 + 1/Sqrt[2]]*x^2) + Sqrt[1 + 1/Sqrt[2]]*Sqrt[1 + x^3])/(x*(1
+ x^3)^(1/4))] + Sqrt[(2 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])] +
Sqrt[(2 - Sqrt[2])/2]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]

________________________________________________________________________________________

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^3+4)*(x^4+x^3+1)/(x^3+1)^(1/4)/(x^8+x^6+2*x^3+1),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+4)*(x^4+x^3+1)/(x^3+1)^(1/4)/(x^8+x^6+2*x^3+1),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 + 1)*(x^3 + 4)/((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 1)^(1/4)), x)

________________________________________________________________________________________

maple [C]  time = 14.73, size = 693, normalized size = 2.52

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{7} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{11} x^{4}+4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{3}+16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{3}+64 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right ) x^{2}+128 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{3}+4}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+16\right )^{9} x^{4}+4 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}+16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{2}-32 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5}+64 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 x^{3} \RootOf \left (\textit {\_Z}^{8}+16\right )-16 \RootOf \left (\textit {\_Z}^{8}+16\right )}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{3}-4}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{11} x^{4}-4 x^{4} \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}-16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{6} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{3}-16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{7}+16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{3}+64 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right ) x^{2}+128 \left (x^{3}+1\right )^{\frac {3}{4}} x +16 \RootOf \left (\textit {\_Z}^{8}+16\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}+4 x^{3}+4}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{5} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+16\right )^{9} x^{4}+4 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{7} x^{2}-4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{4}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5} x^{3}-16 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{8}+16\right )^{3} x^{2}+32 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+16\right )^{2} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}+16\right )^{5}+64 \left (x^{3}+1\right )^{\frac {3}{4}} x -16 x^{3} \RootOf \left (\textit {\_Z}^{8}+16\right )-16 \RootOf \left (\textit {\_Z}^{8}+16\right )}{\RootOf \left (\textit {\_Z}^{8}+16\right )^{4} x^{4}-4 x^{3}-4}\right )}{8}\)	\(693\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+4)*(x^4+x^3+1)/(x^3+1)^(1/4)/(x^8+x^6+2*x^3+1),x,method=_RETURNVERBOSE)

[Out]

-1/16*RootOf(_Z^8+16)^7*ln(-(RootOf(_Z^8+16)^11*x^4+4*x^4*RootOf(_Z^8+16)^7+16*RootOf(_Z^8+16)^6*(x^3+1)^(1/4)
*x^3-4*RootOf(_Z^8+16)^7*x^3+16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^5*x^2-4*RootOf(_Z^8+16)^7-16*RootOf(_Z^8+16)^3*x
^3+64*(x^3+1)^(1/2)*RootOf(_Z^8+16)*x^2+128*(x^3+1)^(3/4)*x-16*RootOf(_Z^8+16)^3)/(RootOf(_Z^8+16)^4*x^4+4*x^3
+4))-1/2*RootOf(_Z^8+16)*ln(-(-RootOf(_Z^8+16)^9*x^4+4*(x^3+1)^(1/2)*RootOf(_Z^8+16)^7*x^2-4*RootOf(_Z^8+16)^5
*x^4-4*RootOf(_Z^8+16)^5*x^3+16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^3*x^2-32*(x^3+1)^(1/4)*RootOf(_Z^8+16)^2*x^3-4*R
ootOf(_Z^8+16)^5+64*(x^3+1)^(3/4)*x-16*x^3*RootOf(_Z^8+16)-16*RootOf(_Z^8+16))/(RootOf(_Z^8+16)^4*x^4-4*x^3-4)
)+1/4*RootOf(_Z^8+16)^3*ln(-(RootOf(_Z^8+16)^11*x^4-4*x^4*RootOf(_Z^8+16)^7-16*RootOf(_Z^8+16)^6*(x^3+1)^(1/4)
*x^3-4*RootOf(_Z^8+16)^7*x^3-16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^5*x^2-4*RootOf(_Z^8+16)^7+16*RootOf(_Z^8+16)^3*x
^3+64*(x^3+1)^(1/2)*RootOf(_Z^8+16)*x^2+128*(x^3+1)^(3/4)*x+16*RootOf(_Z^8+16)^3)/(RootOf(_Z^8+16)^4*x^4+4*x^3
+4))+1/8*RootOf(_Z^8+16)^5*ln(-(RootOf(_Z^8+16)^9*x^4+4*(x^3+1)^(1/2)*RootOf(_Z^8+16)^7*x^2-4*RootOf(_Z^8+16)^
5*x^4+4*RootOf(_Z^8+16)^5*x^3-16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^3*x^2+32*(x^3+1)^(1/4)*RootOf(_Z^8+16)^2*x^3+4*
RootOf(_Z^8+16)^5+64*(x^3+1)^(3/4)*x-16*x^3*RootOf(_Z^8+16)-16*RootOf(_Z^8+16))/(RootOf(_Z^8+16)^4*x^4-4*x^3-4
))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+4)*(x^4+x^3+1)/(x^3+1)^(1/4)/(x^8+x^6+2*x^3+1),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 + 1)*(x^3 + 4)/((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 1)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^3+4\right )\,\left (x^4+x^3+1\right )}{{\left (x^3+1\right )}^{1/4}\,\left (x^8+x^6+2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 4)*(x^3 + x^4 + 1))/((x^3 + 1)^(1/4)*(2*x^3 + x^6 + x^8 + 1)),x)

[Out]

int(((x^3 + 4)*(x^3 + x^4 + 1))/((x^3 + 1)^(1/4)*(2*x^3 + x^6 + x^8 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3} + 4\right ) \left (x^{4} + x^{3} + 1\right )}{\sqrt [4]{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{8} + x^{6} + 2 x^{3} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+4)*(x**4+x**3+1)/(x**3+1)**(1/4)/(x**8+x**6+2*x**3+1),x)

[Out]

Integral((x**3 + 4)*(x**4 + x**3 + 1)/(((x + 1)*(x**2 - x + 1))**(1/4)*(x**8 + x**6 + 2*x**3 + 1)), x)

________________________________________________________________________________________