\(\int \frac {1}{x^4 (a+c x^4)^2} \, dx\) [85]

Optimal result

Integrand size = 13, antiderivative size = 162 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {1}{3 a^2 x^3}-\frac {c x}{4 a^2 \left (a+c x^4\right )}+\frac {7 c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {7 c^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{8 \sqrt {2} a^{11/4}} \] Output:

-1/3/a^2/x^3-1/4*c*x/a^2/(c*x^4+a)-7/16*c^(3/4)*arctan(-1+2^(1/2)*c^(1/4)* 
x/a^(1/4))*2^(1/2)/a^(11/4)-7/16*c^(3/4)*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4 
))*2^(1/2)/a^(11/4)-7/16*c^(3/4)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2 
)+c^(1/2)*x^2))*2^(1/2)/a^(11/4)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {-\frac {32 a^{3/4}}{x^3}-\frac {24 a^{3/4} c x}{a+c x^4}+42 \sqrt {2} c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-42 \sqrt {2} c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+21 \sqrt {2} c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-21 \sqrt {2} c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{96 a^{11/4}} \] Input:

Integrate[1/(x^4*(a + c*x^4)^2),x]
 

Output:

((-32*a^(3/4))/x^3 - (24*a^(3/4)*c*x)/(a + c*x^4) + 42*Sqrt[2]*c^(3/4)*Arc 
Tan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 42*Sqrt[2]*c^(3/4)*ArcTan[1 + (Sqrt 
[2]*c^(1/4)*x)/a^(1/4)] + 21*Sqrt[2]*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4) 
*c^(1/4)*x + Sqrt[c]*x^2] - 21*Sqrt[2]*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/ 
4)*c^(1/4)*x + Sqrt[c]*x^2])/(96*a^(11/4))
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.51, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {819, 847, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {7 \int \frac {1}{x^4 \left (c x^4+a\right )}dx}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {7 \left (-\frac {c \int \frac {1}{c x^4+a}dx}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {7 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+c x^4\right )}\)

Input:

Int[1/(x^4*(a + c*x^4)^2),x]
 

Output:

1/(4*a*x^3*(a + c*x^4)) + (7*(-1/3*1/(a*x^3) - (c*((-(ArcTan[1 - (Sqrt[2]* 
c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/ 
4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] 
- Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log 
[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(2*Sqrt[2]*a^(1/4)*c^( 
1/4)))/(2*Sqrt[a])))/a))/(4*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [C] (verified)

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

Time = 0.47 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.47

method	result	size
risch	\(\frac {-\frac {7 c \,x^{4}}{12 a^{2}}-\frac {1}{3 a}}{x^{3} \left (c \,x^{4}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{11}-4 c^{3}\right ) x -a^{3} c^{2} \textit {\_R} \right )\right )}{16}\)	\(76\)
default	\(-\frac {1}{3 a^{2} x^{3}}-\frac {c \left (\frac {x}{4 c \,x^{4}+4 a}+\frac {7 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}\)	\(130\)

Input:

int(1/x^4/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

(-7/12/a^2*c*x^4-1/3/a)/x^3/(c*x^4+a)+7/16*sum(_R*ln((-5*_R^4*a^11-4*c^3)* 
x-a^3*c^2*_R),_R=RootOf(_Z^4*a^11+c^3))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {28 \, c x^{4} + 21 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 21 \, {\left (i \, a^{2} c x^{7} + i \, a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 i \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 21 \, {\left (-i \, a^{2} c x^{7} - i \, a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 i \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) - 21 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 \, a^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, c x\right ) + 16 \, a}{48 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} \] Input:

integrate(1/x^4/(c*x^4+a)^2,x, algorithm="fricas")
 

Output:

-1/48*(28*c*x^4 + 21*(a^2*c*x^7 + a^3*x^3)*(-c^3/a^11)^(1/4)*log(7*a^3*(-c 
^3/a^11)^(1/4) + 7*c*x) + 21*(I*a^2*c*x^7 + I*a^3*x^3)*(-c^3/a^11)^(1/4)*l 
og(7*I*a^3*(-c^3/a^11)^(1/4) + 7*c*x) + 21*(-I*a^2*c*x^7 - I*a^3*x^3)*(-c^ 
3/a^11)^(1/4)*log(-7*I*a^3*(-c^3/a^11)^(1/4) + 7*c*x) - 21*(a^2*c*x^7 + a^ 
3*x^3)*(-c^3/a^11)^(1/4)*log(-7*a^3*(-c^3/a^11)^(1/4) + 7*c*x) + 16*a)/(a^ 
2*c*x^7 + a^3*x^3)
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.36 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {- 4 a - 7 c x^{4}}{12 a^{3} x^{3} + 12 a^{2} c x^{7}} + \operatorname {RootSum} {\left (65536 t^{4} a^{11} + 2401 c^{3}, \left ( t \mapsto t \log {\left (- \frac {16 t a^{3}}{7 c} + x \right )} \right )\right )} \] Input:

integrate(1/x**4/(c*x**4+a)**2,x)
                                                                                    
                                                                                    
 

Output:

(-4*a - 7*c*x**4)/(12*a**3*x**3 + 12*a**2*c*x**7) + RootSum(65536*_t**4*a* 
*11 + 2401*c**3, Lambda(_t, _t*log(-16*_t*a**3/(7*c) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {7 \, c x^{4} + 4 \, a}{12 \, {\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} - \frac {7 \, {\left (\frac {2 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )}}{32 \, a^{2}} \] Input:

integrate(1/x^4/(c*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/12*(7*c*x^4 + 4*a)/(a^2*c*x^7 + a^3*x^3) - 7/32*(2*sqrt(2)*c*arctan(1/2 
*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(s 
qrt(a)*sqrt(sqrt(a)*sqrt(c))) + 2*sqrt(2)*c*arctan(1/2*sqrt(2)*(2*sqrt(c)* 
x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)* 
sqrt(c))) + sqrt(2)*c^(3/4)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + 
sqrt(a))/a^(3/4) - sqrt(2)*c^(3/4)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/ 
4)*x + sqrt(a))/a^(3/4))/a^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=-\frac {c x}{4 \, {\left (c x^{4} + a\right )} a^{2}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{3}} + \frac {7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{3}} - \frac {1}{3 \, a^{2} x^{3}} \] Input:

integrate(1/x^4/(c*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/4*c*x/((c*x^4 + a)*a^2) - 7/16*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2) 
*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/a^3 - 7/16*sqrt(2)*(a*c^3)^(1/4) 
*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/a^3 - 7/32*sq 
rt(2)*(a*c^3)^(1/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/a^3 + 7/3 
2*sqrt(2)*(a*c^3)^(1/4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/a^3 - 
 1/3/(a^2*x^3)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {7\,{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{8\,a^{11/4}}-\frac {\frac {1}{3\,a}+\frac {7\,c\,x^4}{12\,a^2}}{c\,x^7+a\,x^3}+\frac {7\,{\left (-c\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{8\,a^{11/4}} \] Input:

int(1/(x^4*(a + c*x^4)^2),x)
 

Output:

(7*(-c)^(3/4)*atan(((-c)^(1/4)*x)/a^(1/4)))/(8*a^(11/4)) - (1/(3*a) + (7*c 
*x^4)/(12*a^2))/(a*x^3 + c*x^7) + (7*(-c)^(3/4)*atanh(((-c)^(1/4)*x)/a^(1/ 
4)))/(8*a^(11/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^4 \left (a+c x^4\right )^2} \, dx=\frac {42 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+42 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{7}-42 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-42 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{7}+21 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{3}+21 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{7}-21 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{3}-21 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{7}-32 a^{2}-56 a c \,x^{4}}{96 a^{3} x^{3} \left (c \,x^{4}+a \right )} \] Input:

int(1/x^4/(c*x^4+a)^2,x)
 

Output:

(42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)* 
x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*x**3 + 42*c**(3/4)*a**(1/4)*sqrt(2)*atan 
((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c* 
x**7 - 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sq 
rt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*x**3 - 42*c**(3/4)*a**(1/4)*sqrt(2 
)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2 
)))*c*x**7 + 21*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2) 
*x + sqrt(a) + sqrt(c)*x**2)*a*x**3 + 21*c**(3/4)*a**(1/4)*sqrt(2)*log( - 
c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c*x**7 - 21*c**(3/4) 
*a**(1/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2 
)*a*x**3 - 21*c**(3/4)*a**(1/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + 
sqrt(a) + sqrt(c)*x**2)*c*x**7 - 32*a**2 - 56*a*c*x**4)/(96*a**3*x**3*(a + 
 c*x**4))