\(\int \frac {1}{x^3 (a+b x^4) (c+d x^4)} \, dx\) [204]

Optimal result

Integrand size = 22, antiderivative size = 92 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {1}{2 a c x^2}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)}+\frac {d^{3/2} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)} \] Output:

-1/2/a/c/x^2-1/2*b^(3/2)*arctan(b^(1/2)*x^2/a^(1/2))/a^(3/2)/(-a*d+b*c)+1/ 
2*d^(3/2)*arctan(d^(1/2)*x^2/c^(1/2))/c^(3/2)/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.84 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\frac {b}{a}-\frac {d}{c}-\frac {b^{3/2} x^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/2}}-\frac {b^{3/2} x^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/2}}+\frac {d^{3/2} x^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{3/2}}+\frac {d^{3/2} x^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{3/2}}}{2 (-b c+a d) x^2} \] Input:

Integrate[1/(x^3*(a + b*x^4)*(c + d*x^4)),x]
 

Output:

(b/a - d/c - (b^(3/2)*x^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/2) 
 - (b^(3/2)*x^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/2) + (d^(3/2 
)*x^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2) + (d^(3/2)*x^2*ArcT 
an[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2))/(2*(-(b*c) + a*d)*x^2)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {965, 382, 25, 397, 218}

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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 382

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {a d^{3/2} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}}{a c}-\frac {1}{a c x^2}\right )\)

Input:

Int[1/(x^3*(a + b*x^4)*(c + d*x^4)),x]
 

Output:

(-(1/(a*c*x^2)) - ((b^(3/2)*c*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*(b*c 
 - a*d)) - (a*d^(3/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)) 
)/(a*c))/2
 

Defintions of rubi rules used

rule 25

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

rule 218

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

rule 382

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {b^{2} \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 a \left (a d -c b \right ) \sqrt {a b}}-\frac {1}{2 a c \,x^{2}}-\frac {d^{2} \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 c \left (a d -c b \right ) \sqrt {c d}}\)	\(81\)
risch	\(-\frac {1}{2 a c \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{2} a^{5}-2 a^{4} b c d +c^{2} a^{3} b^{2}\right ) \textit {\_Z}^{2}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-5 c^{3} a^{7} d^{4}+18 c^{4} a^{6} b \,d^{3}-26 a^{5} c^{5} b^{2} d^{2}+18 c^{6} a^{4} b^{3} d -5 c^{7} a^{3} b^{4}\right ) \textit {\_R}^{4}+\left (-4 a^{5} d^{5}+8 a^{4} b c \,d^{4}-5 c^{2} a^{3} b^{2} d^{3}-5 c^{3} a^{2} b^{3} d^{2}+8 c^{4} a \,b^{4} d -4 c^{5} b^{5}\right ) \textit {\_R}^{2}-4 b^{3} d^{3}\right ) x^{2}+\left (-a^{6} c^{2} d^{4}+a^{5} b \,c^{3} d^{3}+a^{3} b^{3} c^{5} d -a^{2} b^{4} c^{6}\right ) \textit {\_R}^{3}\right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{2} c^{3} a^{2}-2 a b \,c^{4} d +b^{2} c^{5}\right ) \textit {\_Z}^{2}+d^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-5 c^{3} a^{7} d^{4}+18 c^{4} a^{6} b \,d^{3}-26 a^{5} c^{5} b^{2} d^{2}+18 c^{6} a^{4} b^{3} d -5 c^{7} a^{3} b^{4}\right ) \textit {\_R}^{4}+\left (-4 a^{5} d^{5}+8 a^{4} b c \,d^{4}-5 c^{2} a^{3} b^{2} d^{3}-5 c^{3} a^{2} b^{3} d^{2}+8 c^{4} a \,b^{4} d -4 c^{5} b^{5}\right ) \textit {\_R}^{2}-4 b^{3} d^{3}\right ) x^{2}+\left (-a^{6} c^{2} d^{4}+a^{5} b \,c^{3} d^{3}+a^{3} b^{3} c^{5} d -a^{2} b^{4} c^{6}\right ) \textit {\_R}^{3}\right )\right )}{4}\)	\(493\)

Input:

int(1/x^3/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

1/2*b^2/a/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))-1/2/a/c/x^2-1/2* 
d^2/c/(a*d-b*c)/(c*d)^(1/2)*arctan(d*x^2/(c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 416, normalized size of antiderivative = 4.52 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [-\frac {b c x^{2} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} + 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + a d x^{2} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} - 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) + 2 \, b c - 2 \, a d}{4 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}}, \frac {2 \, a d x^{2} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) - b c x^{2} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} + 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) - 2 \, b c + 2 \, a d}{4 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}}, -\frac {2 \, b c x^{2} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) + a d x^{2} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} - 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) + 2 \, b c - 2 \, a d}{4 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}}, -\frac {b c x^{2} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) - a d x^{2} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) + b c - a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}}\right ] \] Input:

integrate(1/x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
 

Output:

[-1/4*(b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a) 
) + a*d*x^2*sqrt(-d/c)*log((d*x^4 - 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) + 
 2*b*c - 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), 1/4*(2*a*d*x^2*sqrt(d/c)*arctan 
(x^2*sqrt(d/c)) - b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/ 
(b*x^4 + a)) - 2*b*c + 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), -1/4*(2*b*c*x^2*s 
qrt(b/a)*arctan(x^2*sqrt(b/a)) + a*d*x^2*sqrt(-d/c)*log((d*x^4 - 2*c*x^2*s 
qrt(-d/c) - c)/(d*x^4 + c)) + 2*b*c - 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), -1 
/2*(b*c*x^2*sqrt(b/a)*arctan(x^2*sqrt(b/a)) - a*d*x^2*sqrt(d/c)*arctan(x^2 
*sqrt(d/c)) + b*c - a*d)/((a*b*c^2 - a^2*c*d)*x^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (76) = 152\).

Time = 72.16 (sec) , antiderivative size = 1103, normalized size of antiderivative = 11.99 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x**3/(b*x**4+a)/(d*x**4+c),x)
 

Output:

-sqrt(-b**3/a**3)*log(x**2 + (-a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b 
*c)**3 + 2*a**6*b*c**4*d**3*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**5*b* 
*2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqrt(-b**3/a** 
3)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a 
**3*b**4*c**7*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-b**3/a* 
*3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(4*(a* 
d - b*c)) + sqrt(-b**3/a**3)*log(x**2 + (a**7*c**3*d**4*(-b**3/a**3)**(3/2 
)/(a*d - b*c)**3 - 2*a**6*b*c**4*d**3*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + 
 2*a**5*b**2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + a**5*d**5*sqrt 
(-b**3/a**3)/(a*d - b*c) - 2*a**4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b 
*c)**3 + a**3*b**4*c**7*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + b**5*c**5*sqr 
t(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d** 
2))/(4*(a*d - b*c)) - sqrt(-d**3/c**3)*log(x**2 + (-a**7*c**3*d**4*(-d**3/ 
c**3)**(3/2)/(a*d - b*c)**3 + 2*a**6*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d 
- b*c)**3 - 2*a**5*b**2*c**5*d**2*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a** 
5*d**5*sqrt(-d**3/c**3)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/ 
2)/(a*d - b*c)**3 - a**3*b**4*c**7*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - b* 
*5*c**5*sqrt(-d**3/c**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b* 
*4*c**2*d**2))/(4*(a*d - b*c)) + sqrt(-d**3/c**3)*log(x**2 + (a**7*c**3*d* 
*4*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - 2*a**6*b*c**4*d**3*(-d**3/c**3)...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {b^{2} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{2 \, a c x^{2}} \] Input:

integrate(1/x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
 

Output:

-1/2*b^2*arctan(b*x^2/sqrt(a*b))/((a*b*c - a^2*d)*sqrt(a*b)) + 1/2*d^2*arc 
tan(d*x^2/sqrt(c*d))/((b*c^2 - a*c*d)*sqrt(c*d)) - 1/2/(a*c*x^2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {b^{2} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{2 \, a c x^{2}} \] Input:

integrate(1/x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
 

Output:

-1/2*b^2*arctan(b*x^2/sqrt(a*b))/((a*b*c - a^2*d)*sqrt(a*b)) + 1/2*d^2*arc 
tan(d*x^2/sqrt(c*d))/((b*c^2 - a*c*d)*sqrt(c*d)) - 1/2/(a*c*x^2)
 

Mupad [B] (verification not implemented)

Time = 4.73 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.85 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\ln \left (c^3\,x^2\,{\left (-a^3\,b^3\right )}^{3/2}-a^8\,b\,d^3+a^5\,b^4\,c^3+a^6\,d^3\,x^2\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{4\,a^4\,d-4\,a^3\,b\,c}-\frac {\ln \left (c^3\,x^2\,{\left (-a^3\,b^3\right )}^{3/2}+a^8\,b\,d^3-a^5\,b^4\,c^3+a^6\,d^3\,x^2\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{4\,\left (a^4\,d-a^3\,b\,c\right )}-\frac {1}{2\,a\,c\,x^2}-\frac {\ln \left (a^3\,x^2\,{\left (-c^3\,d^3\right )}^{3/2}+b^3\,c^8\,d-a^3\,c^5\,d^4+b^3\,c^6\,x^2\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{4\,\left (b\,c^4-a\,c^3\,d\right )}+\frac {\ln \left (a^3\,x^2\,{\left (-c^3\,d^3\right )}^{3/2}-b^3\,c^8\,d+a^3\,c^5\,d^4+b^3\,c^6\,x^2\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{4\,b\,c^4-4\,a\,c^3\,d} \] Input:

int(1/(x^3*(a + b*x^4)*(c + d*x^4)),x)
 

Output:

(log(c^3*x^2*(-a^3*b^3)^(3/2) - a^8*b*d^3 + a^5*b^4*c^3 + a^6*d^3*x^2*(-a^ 
3*b^3)^(1/2))*(-a^3*b^3)^(1/2))/(4*a^4*d - 4*a^3*b*c) - (log(c^3*x^2*(-a^3 
*b^3)^(3/2) + a^8*b*d^3 - a^5*b^4*c^3 + a^6*d^3*x^2*(-a^3*b^3)^(1/2))*(-a^ 
3*b^3)^(1/2))/(4*(a^4*d - a^3*b*c)) - 1/(2*a*c*x^2) - (log(a^3*x^2*(-c^3*d 
^3)^(3/2) + b^3*c^8*d - a^3*c^5*d^4 + b^3*c^6*x^2*(-c^3*d^3)^(1/2))*(-c^3* 
d^3)^(1/2))/(4*(b*c^4 - a*c^3*d)) + (log(a^3*x^2*(-c^3*d^3)^(3/2) - b^3*c^ 
8*d + a^3*c^5*d^4 + b^3*c^6*x^2*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/(4*b*c 
^4 - 4*a*c^3*d)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) b \,c^{2} x^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) b \,c^{2} x^{2}+\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{2} d \,x^{2}+\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{2} d \,x^{2}-a^{2} c d +a b \,c^{2}}{2 a^{2} c^{2} x^{2} \left (a d -b c \right )} \] Input:

int(1/x^3/(b*x^4+a)/(d*x^4+c),x)
 

Output:

( - sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/ 
4)*a**(1/4)*sqrt(2)))*b*c**2*x**2 - sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4 
)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c**2*x**2 + sqrt(d 
)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4 
)*sqrt(2)))*a**2*d*x**2 + sqrt(d)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt(2) 
+ 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d*x**2 - a**2*c*d + a*b*c 
**2)/(2*a**2*c**2*x**2*(a*d - b*c))