\(\int \frac {(4+x^3) (1+x^3+x^4)}{\sqrt [4]{1+x^3} (1+2 x^3+x^6+x^8)} \, dx\) [2808]

Optimal result

Integrand size = 38, antiderivative size = 275 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=-\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\frac {\sqrt {2-\sqrt {2}} x^2}{-2+\sqrt {2}}-\frac {\sqrt {2-\sqrt {2}} \sqrt {1+x^3}}{-2+\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {-\frac {x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {1+x^3}}{\sqrt {2+\sqrt {2}}}}{x \sqrt [4]{1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[Out]

-1/2*(4+2*2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)*x^2/(-2+2^(1/2))-(2-2^(1/2))^(1/2)*(x^3+1)^(1/2)/(-2+2^(1/2
)))/x/(x^3+1)^(1/4))-1/2*(4-2*2^(1/2))^(1/2)*arctan((-x^2/(2+2^(1/2))^(1/2)+(x^3+1)^(1/2)/(2+2^(1/2))^(1/2))/x
/(x^3+1)^(1/4))+1/2*(4+2*2^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)*x*(x^3+1)^(1/4)/(x^2+(x^3+1)^(1/2)))+1/2*(4-
2*2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*x*(x^3+1)^(1/4)/(x^2+(x^3+1)^(1/2)))

Rubi [F]

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

[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*} \text {integral}& = \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 [A] (verified)

Time = 2.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.80 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\frac {-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (-x^2+\sqrt {1+x^3}\right )}{x \sqrt [4]{1+x^3}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (-x^2+\sqrt {1+x^3}\right )}{x \sqrt [4]{1+x^3}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}} \]

[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]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.16 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.52

method	result	size
trager	\(\text {Expression too large to display}\)	\(693\)

[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*RootOf(_Z^8+16)^7*x^4+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))+1/4*RootOf(_Z^8+16)^3*ln(-(RootOf(_Z^8+16)^11*x^4-4*RootOf(_Z^8+16)^7*x^4-16*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*
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
))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 47.89 (sec) , antiderivative size = 1537, normalized size of antiderivative = 5.59 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\text {Too large to display} \]

[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]

-1/4*sqrt(2)*(-1)^(1/8)*log(8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(x^6 - x^5 - x^2) + (-1)^(3/8)*(x^6 + x^5 +
x^2)) + 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(x^8 + 2*x^7 - x^6 + 2*x^4 - 2*x^3 - 1) -
(-1)^(1/8)*(x^8 - 2*x^7 - x^6 - 2*x^4 - 2*x^3 - 1)) - 4*((-1)^(1/4)*x^7 - (-1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(
1/4))/(x^8 + x^6 + 2*x^3 + 1)) + 1/4*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(x^6 - x^5
- x^2) + (-1)^(3/8)*(x^6 + x^5 + x^2)) - 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(x^8 + 2
*x^7 - x^6 + 2*x^4 - 2*x^3 - 1) - (-1)^(1/8)*(x^8 - 2*x^7 - x^6 - 2*x^4 - 2*x^3 - 1)) + 4*((-1)^(1/4)*x^7 - (-
1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^8 + x^6 + 2*x^3 + 1)) - 1/4*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*s
qrt(x^3 + 1)*((-1)^(7/8)*(I*x^6 - I*x^5 - I*x^2) + (-1)^(3/8)*(I*x^6 + I*x^5 + I*x^2)) - 4*(x^5 - I*x^4 - I*x)
*(x^3 + 1)^(3/4) + sqrt(2)*((-1)^(5/8)*(I*x^8 + 2*I*x^7 - I*x^6 + 2*I*x^4 - 2*I*x^3 - I) + (-1)^(1/8)*(-I*x^8
+ 2*I*x^7 + I*x^6 + 2*I*x^4 + 2*I*x^3 + I)) - 4*((-1)^(1/4)*x^7 - (-1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^
8 + x^6 + 2*x^3 + 1)) + 1/4*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(-I*x^6 + I*x^5 +
I*x^2) + (-1)^(3/8)*(-I*x^6 - I*x^5 - I*x^2)) - 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) + sqrt(2)*((-1)^(5/8)*(
-I*x^8 - 2*I*x^7 + I*x^6 - 2*I*x^4 + 2*I*x^3 + I) + (-1)^(1/8)*(I*x^8 - 2*I*x^7 - I*x^6 - 2*I*x^4 - 2*I*x^3 -
I)) - 4*((-1)^(1/4)*x^7 - (-1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^8 + x^6 + 2*x^3 + 1)) + (1/4*I + 1/4)*(-
1)^(1/8)*log(16*(4*(x^5 + I*x^4 + I*x)*(x^3 + 1)^(3/4) - (-1)^(5/8)*((I + 1)*x^8 + (2*I + 2)*x^7 - (I + 1)*x^6
+ (2*I + 2)*x^4 - (2*I + 2)*x^3 - I - 1) - 2*sqrt(x^3 + 1)*((-1)^(7/8)*(-(I - 1)*x^6 + (I - 1)*x^5 + (I - 1)*
x^2) + (-1)^(3/8)*((I - 1)*x^6 + (I - 1)*x^5 + (I - 1)*x^2)) - 4*(I*(-1)^(1/4)*x^7 + (-1)^(3/4)*(I*x^6 + I*x^3
))*(x^3 + 1)^(1/4) - (-1)^(1/8)*((I + 1)*x^8 - (2*I + 2)*x^7 - (I + 1)*x^6 - (2*I + 2)*x^4 - (2*I + 2)*x^3 - I
- 1))/(x^8 + x^6 + 2*x^3 + 1)) - (1/4*I - 1/4)*(-1)^(1/8)*log(16*(4*(x^5 + I*x^4 + I*x)*(x^3 + 1)^(3/4) - (-1
)^(5/8)*(-(I - 1)*x^8 - (2*I - 2)*x^7 + (I - 1)*x^6 - (2*I - 2)*x^4 + (2*I - 2)*x^3 + I - 1) - 2*sqrt(x^3 + 1)
*((-1)^(7/8)*((I + 1)*x^6 - (I + 1)*x^5 - (I + 1)*x^2) + (-1)^(3/8)*(-(I + 1)*x^6 - (I + 1)*x^5 - (I + 1)*x^2)
) - 4*(-I*(-1)^(1/4)*x^7 + (-1)^(3/4)*(-I*x^6 - I*x^3))*(x^3 + 1)^(1/4) - (-1)^(1/8)*(-(I - 1)*x^8 + (2*I - 2)
*x^7 + (I - 1)*x^6 + (2*I - 2)*x^4 + (2*I - 2)*x^3 + I - 1))/(x^8 + x^6 + 2*x^3 + 1)) + (1/4*I - 1/4)*(-1)^(1/
8)*log(16*(4*(x^5 + I*x^4 + I*x)*(x^3 + 1)^(3/4) - (-1)^(5/8)*((I - 1)*x^8 + (2*I - 2)*x^7 - (I - 1)*x^6 + (2*
I - 2)*x^4 - (2*I - 2)*x^3 - I + 1) - 2*sqrt(x^3 + 1)*((-1)^(7/8)*(-(I + 1)*x^6 + (I + 1)*x^5 + (I + 1)*x^2) +
(-1)^(3/8)*((I + 1)*x^6 + (I + 1)*x^5 + (I + 1)*x^2)) - 4*(-I*(-1)^(1/4)*x^7 + (-1)^(3/4)*(-I*x^6 - I*x^3))*(
x^3 + 1)^(1/4) - (-1)^(1/8)*((I - 1)*x^8 - (2*I - 2)*x^7 - (I - 1)*x^6 - (2*I - 2)*x^4 - (2*I - 2)*x^3 - I + 1
))/(x^8 + x^6 + 2*x^3 + 1)) - (1/4*I + 1/4)*(-1)^(1/8)*log(16*(4*(x^5 + I*x^4 + I*x)*(x^3 + 1)^(3/4) - (-1)^(5
/8)*(-(I + 1)*x^8 - (2*I + 2)*x^7 + (I + 1)*x^6 - (2*I + 2)*x^4 + (2*I + 2)*x^3 + I + 1) - 2*sqrt(x^3 + 1)*((-
1)^(7/8)*((I - 1)*x^6 - (I - 1)*x^5 - (I - 1)*x^2) + (-1)^(3/8)*(-(I - 1)*x^6 - (I - 1)*x^5 - (I - 1)*x^2)) -
4*(I*(-1)^(1/4)*x^7 + (-1)^(3/4)*(I*x^6 + I*x^3))*(x^3 + 1)^(1/4) - (-1)^(1/8)*(-(I + 1)*x^8 + (2*I + 2)*x^7 +
(I + 1)*x^6 + (2*I + 2)*x^4 + (2*I + 2)*x^3 + I + 1))/(x^8 + x^6 + 2*x^3 + 1))

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\int \frac {\left (x^3+4\right )\,\left (x^4+x^3+1\right )}{{\left (x^3+1\right )}^{1/4}\,\left (x^8+x^6+2\,x^3+1\right )} \,d x \]

[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)