\(\int \frac {1-x^4}{(1+x^4) \sqrt [4]{x^3+x^5}} \, dx\) [2089]

Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{\sqrt [8]{2}}-\frac {\arctan \left (\frac {2^{5/8} x \sqrt [4]{x^3+x^5}}{\sqrt [4]{2} x^2-\sqrt {x^3+x^5}}\right )}{2^{5/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{\sqrt [8]{2}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^3+x^5}}{2^{5/8}}}{x \sqrt [4]{x^3+x^5}}\right )}{2^{5/8}} \]

[Out]

1/2*arctan(2^(1/8)*x/(x^5+x^3)^(1/4))*2^(7/8)-1/2*arctan(2^(5/8)*x*(x^5+x^3)^(1/4)/(x^2*2^(1/4)-(x^5+x^3)^(1/2
)))*2^(3/8)+1/2*arctanh(2^(1/8)*x/(x^5+x^3)^(1/4))*2^(7/8)+1/2*arctanh((1/2*x^2*2^(5/8)+1/2*(x^5+x^3)^(1/2)*2^
(3/8))/x/(x^5+x^3)^(1/4))*2^(3/8)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2081, 1600, 6847, 6857, 440} \[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {(2-2 i) x \sqrt [4]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},-i x^2,-x^2\right )}{\sqrt [4]{x^5+x^3}}+\frac {(2+2 i) x \sqrt [4]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},i x^2,-x^2\right )}{\sqrt [4]{x^5+x^3}} \]

[In]

Int[(1 - x^4)/((1 + x^4)*(x^3 + x^5)^(1/4)),x]

[Out]

((2 - 2*I)*x*(1 + x^2)^(1/4)*AppellF1[1/8, 1, -3/4, 9/8, (-I)*x^2, -x^2])/(x^3 + x^5)^(1/4) + ((2 + 2*I)*x*(1
+ x^2)^(1/4)*AppellF1[1/8, 1, -3/4, 9/8, I*x^2, -x^2])/(x^3 + x^5)^(1/4)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {1-x^4}{x^{3/4} \sqrt [4]{1+x^2} \left (1+x^4\right )} \, dx}{\sqrt [4]{x^3+x^5}} \\ & = \frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {\left (1-x^2\right ) \left (1+x^2\right )^{3/4}}{x^{3/4} \left (1+x^4\right )} \, dx}{\sqrt [4]{x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1-x^8\right ) \left (1+x^8\right )^{3/4}}{1+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+x^8\right )^{3/4}}{i-x^8}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+x^8\right )^{3/4}}{i+x^8}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = -\frac {\left ((2-2 i) x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^8\right )^{3/4}}{i+x^8} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left ((2+2 i) x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^8\right )^{3/4}}{i-x^8} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {(2-2 i) x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},-i x^2,-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {(2+2 i) x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},i x^2,-x^2\right )}{\sqrt [4]{x^3+x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.20 \[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {x^{3/4} \sqrt [4]{1+x^2} \left (\sqrt {2} \arctan \left (\frac {\sqrt [8]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\arctan \left (\frac {2^{5/8} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt [4]{2} \sqrt {x}-\sqrt {1+x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [8]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/8} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+2^{3/4} \sqrt {1+x^2}}\right )\right )}{2^{5/8} \sqrt [4]{x^3+x^5}} \]

[In]

Integrate[(1 - x^4)/((1 + x^4)*(x^3 + x^5)^(1/4)),x]

[Out]

(x^(3/4)*(1 + x^2)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/8)*x^(1/4))/(1 + x^2)^(1/4)] - ArcTan[(2^(5/8)*x^(1/4)*(1 + x^2
)^(1/4))/(2^(1/4)*Sqrt[x] - Sqrt[1 + x^2])] + Sqrt[2]*ArcTanh[(2^(1/8)*x^(1/4))/(1 + x^2)^(1/4)] + ArcTanh[(2*
2^(3/8)*x^(1/4)*(1 + x^2)^(1/4))/(2*Sqrt[x] + 2^(3/4)*Sqrt[1 + x^2])]))/(2^(5/8)*(x^3 + x^5)^(1/4))

Maple [C] (verified)

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

Time = 26.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.25

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{2}\)	\(37\)
trager	\(\text {Expression too large to display}\)	\(1752\)

[In]

int((-x^4+1)/(x^4+1)/(x^5+x^3)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/2*sum(ln((-_R*x+(x^3*(x^2+1))^(1/4))/x)/_R,_R=RootOf(_Z^8-2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 14.48 (sec) , antiderivative size = 1640, normalized size of antiderivative = 10.86 \[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\text {Too large to display} \]

[In]

integrate((-x^4+1)/(x^4+1)/(x^5+x^3)^(1/4),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=- \int \frac {x^{4}}{x^{4} \sqrt [4]{x^{5} + x^{3}} + \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \left (- \frac {1}{x^{4} \sqrt [4]{x^{5} + x^{3}} + \sqrt [4]{x^{5} + x^{3}}}\right )\, dx \]

[In]

integrate((-x**4+1)/(x**4+1)/(x**5+x**3)**(1/4),x)

[Out]

-Integral(x**4/(x**4*(x**5 + x**3)**(1/4) + (x**5 + x**3)**(1/4)), x) - Integral(-1/(x**4*(x**5 + x**3)**(1/4)
+ (x**5 + x**3)**(1/4)), x)

Maxima [F]

\[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((-x^4+1)/(x^4+1)/(x^5+x^3)^(1/4),x, algorithm="maxima")

[Out]

-integrate((x^4 - 1)/((x^5 + x^3)^(1/4)*(x^4 + 1)), x)

Giac [F]

\[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((-x^4+1)/(x^4+1)/(x^5+x^3)^(1/4),x, algorithm="giac")

[Out]

integrate(-(x^4 - 1)/((x^5 + x^3)^(1/4)*(x^4 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=-\int \frac {x^4-1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^4+1\right )} \,d x \]

[In]

int(-(x^4 - 1)/((x^3 + x^5)^(1/4)*(x^4 + 1)),x)

[Out]

-int((x^4 - 1)/((x^3 + x^5)^(1/4)*(x^4 + 1)), x)