3.2.48 \(\int \frac {1}{(d+e x^2) (a+c x^4)^2} \, dx\) [148]

3.2.48.1 Optimal result
3.2.48.2 Mathematica [A] (verified)
3.2.48.3 Rubi [A] (verified)
3.2.48.4 Maple [A] (verified)
3.2.48.5 Fricas [B] (verification not implemented)
3.2.48.6 Sympy [F(-1)]
3.2.48.7 Maxima [F(-2)]
3.2.48.8 Giac [A] (verification not implemented)
3.2.48.9 Mupad [B] (verification not implemented)

3.2.48.1 Optimal result

Integrand size = 19, antiderivative size = 689 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {c x \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{c} e^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{c} e^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{c} e^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{c} e^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )} \]

output
1/4*c*x*(-e*x^2+d)/a/(a*e^2+c*d^2)/(c*x^4+a)+1/4*c^(1/4)*e^2*arctan(-1+c^( 
1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+d*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^( 
1/2)+1/4*c^(1/4)*e^2*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+d*c^( 
1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/8*c^(1/4)*e^2*ln(-a^(1/4)*c^(1/4)* 
x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+d*c^(1/2))/a^(3/4)/(a*e^2+c*d^2) 
^2*2^(1/2)+1/8*c^(1/4)*e^2*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2 
))*(e*a^(1/2)+d*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/16*c^(1/4)*arct 
an(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+3*d*c^(1/2))/a^(7/4)/(a*e^2+c 
*d^2)*2^(1/2)+1/16*c^(1/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2) 
+3*d*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)*2^(1/2)-1/32*c^(1/4)*ln(-a^(1/4)*c^(1/ 
4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+3*d*c^(1/2))/a^(7/4)/(a*e^2+c 
*d^2)*2^(1/2)+1/32*c^(1/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2 
))*(e*a^(1/2)+3*d*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)*2^(1/2)+e^(7/2)*arctan(x* 
e^(1/2)/d^(1/2))/(a*e^2+c*d^2)^2/d^(1/2)
 
3.2.48.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 c \left (c d^2+a e^2\right ) x \left (d-e x^2\right )}{a \left (a+c x^4\right )}+\frac {32 e^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {2 \sqrt {2} \sqrt [4]{c} \left (-3 c^{3/2} d^3+\sqrt {a} c d^2 e-7 a \sqrt {c} d e^2+5 a^{3/2} e^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {2 \sqrt {2} \sqrt [4]{c} \left (-3 c^{3/2} d^3+\sqrt {a} c d^2 e-7 a \sqrt {c} d e^2+5 a^{3/2} e^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{3/2} d^3+\sqrt {a} c d^2 e+7 a \sqrt {c} d e^2+5 a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{3/2} d^3+\sqrt {a} c d^2 e+7 a \sqrt {c} d e^2+5 a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}}{32 \left (c d^2+a e^2\right )^2} \]

input
Integrate[1/((d + e*x^2)*(a + c*x^4)^2),x]
 
output
((8*c*(c*d^2 + a*e^2)*x*(d - e*x^2))/(a*(a + c*x^4)) + (32*e^(7/2)*ArcTan[ 
(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[2]*c^(1/4)*(-3*c^(3/2)*d^3 + Sqrt[ 
a]*c*d^2*e - 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4 
)*x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(1/4)*(-3*c^(3/2)*d^3 + Sqrt[a]*c*d^ 
2*e - 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^ 
(1/4)])/a^(7/4) - (Sqrt[2]*c^(1/4)*(3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 7*a* 
Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + S 
qrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(1/4)*(3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 
 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)* 
x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^2 + a*e^2)^2)
 
3.2.48.3 Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1568, 2009}

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.

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

\(\Big \downarrow \) 1568

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt [4]{c} e^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{c} e^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {c} d-\sqrt {a} e\right )}{8 \sqrt {2} a^{7/4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {c} d-\sqrt {a} e\right )}{8 \sqrt {2} a^{7/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{c} e^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{c} e^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {a} e+3 \sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a} e+3 \sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (a e^2+c d^2\right )}+\frac {e^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (a e^2+c d^2\right )^2}+\frac {c x \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\)

input
Int[1/((d + e*x^2)*(a + c*x^4)^2),x]
 
output
(c*x*(d - e*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(7/2)*ArcTan[(Sqr 
t[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 + a*e^2)^2) - (c^(1/4)*e^2*(Sqrt[c]*d - 
Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c* 
d^2 + a*e^2)^2) - (c^(1/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c 
^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)) + (c^(1/4)*e^2*(Sq 
rt[c]*d - Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a 
^(3/4)*(c*d^2 + a*e^2)^2) + (c^(1/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 + 
(Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)) - (c^(1/ 
4)*e^2*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + S 
qrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) - (c^(1/4)*(3*Sqrt[c]*d 
 + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16* 
Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)) + (c^(1/4)*e^2*(Sqrt[c]*d + Sqrt[a]*e)*Lo 
g[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*( 
c*d^2 + a*e^2)^2) + (c^(1/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[ 
2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2))
 

3.2.48.3.1 Defintions of rubi rules used

rule 1568
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int 
[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, 
p, q}, x] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0])
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 
3.2.48.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.48

method result size
default \(\frac {c \left (\frac {-\frac {e \left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{4 a}+\frac {d \left (a \,e^{2}+c \,d^{2}\right ) x}{4 a}}{c \,x^{4}+a}+\frac {\frac {\left (7 d \,e^{2} a +3 d^{3} c \right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-5 a \,e^{3}-c \,d^{2} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {e^{4} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e d}}\) \(334\)
risch \(\text {Expression too large to display}\) \(1470\)

input
int(1/(e*x^2+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
 
output
c/(a*e^2+c*d^2)^2*((-1/4*e*(a*e^2+c*d^2)/a*x^3+1/4*d*(a*e^2+c*d^2)/a*x)/(c 
*x^4+a)+1/4/a*(1/8*(7*a*d*e^2+3*c*d^3)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c 
)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2* 
arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/8*(-5 
*a*e^3-c*d^2*e)/c/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c) 
^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1 
/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))))+e^4/(a*e^2+c*d^2)^2/(e*d)^(1 
/2)*arctan(e*x/(e*d)^(1/2))
 
3.2.48.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4934 vs. \(2 (518) = 1036\).

Time = 12.22 (sec) , antiderivative size = 9892, normalized size of antiderivative = 14.36 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

input
integrate(1/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")
 
output
Too large to include
 
3.2.48.6 Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]

input
integrate(1/(e*x**2+d)/(c*x**4+a)**2,x)
 
output
Timed out
 
3.2.48.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]

input
integrate(1/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")
 
output
Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 
3.2.48.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 621, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {e^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {d e}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} a^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} - \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} a^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} - \frac {c e x^{3} - c d x}{4 \, {\left (c x^{4} + a\right )} {\left (a c d^{2} + a^{2} e^{2}\right )}} \]

input
integrate(1/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")
 
output
e^4*arctan(e*x/sqrt(d*e))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(d*e)) 
+ 1/8*(3*(a*c^3)^(1/4)*c^3*d^3 + 7*(a*c^3)^(1/4)*a*c^2*d*e^2 - (a*c^3)^(3/ 
4)*c*d^2*e - 5*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c 
)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^4 + 2*sqrt(2)*a^3*c^3*d^2*e^2 + s 
qrt(2)*a^4*c^2*e^4) + 1/8*(3*(a*c^3)^(1/4)*c^3*d^3 + 7*(a*c^3)^(1/4)*a*c^2 
*d*e^2 - (a*c^3)^(3/4)*c*d^2*e - 5*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2) 
*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^4 + 2*sqrt(2) 
*a^3*c^3*d^2*e^2 + sqrt(2)*a^4*c^2*e^4) + 1/16*(3*(a*c^3)^(1/4)*c^3*d^3 + 
7*(a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^3)^(3/4)*c*d^2*e + 5*(a*c^3)^(3/4)*a*e^ 
3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^4*d^4 + 2*s 
qrt(2)*a^3*c^3*d^2*e^2 + sqrt(2)*a^4*c^2*e^4) - 1/16*(3*(a*c^3)^(1/4)*c^3* 
d^3 + 7*(a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^3)^(3/4)*c*d^2*e + 5*(a*c^3)^(3/4 
)*a*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^4*d^4 
 + 2*sqrt(2)*a^3*c^3*d^2*e^2 + sqrt(2)*a^4*c^2*e^4) - 1/4*(c*e*x^3 - c*d*x 
)/((c*x^4 + a)*(a*c*d^2 + a^2*e^2))
 
3.2.48.9 Mupad [B] (verification not implemented)

Time = 16.36 (sec) , antiderivative size = 17945, normalized size of antiderivative = 26.04 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

input
int(1/((a + c*x^4)^2*(d + e*x^2)),x)
 
output
((c*d*x)/(4*a*(a*e^2 + c*d^2)) - (c*e*x^3)/(4*a*(a*e^2 + c*d^2)))/(a + c*x 
^4) - atan(((((((65536*a^11*c^4*e^16 - 12288*a^4*c^11*d^14*e^2 - 57344*a^5 
*c^10*d^12*e^4 - 36864*a^6*c^9*d^10*e^6 + 245760*a^7*c^8*d^8*e^8 + 634880* 
a^8*c^7*d^6*e^10 + 663552*a^9*c^6*d^4*e^12 + 331776*a^10*c^5*d^2*e^14)/(25 
6*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2 
*d^4*e^4)) - (x*((9*c^3*d^6*(-a^7*c)^(1/2) - 25*a^3*e^6*(-a^7*c)^(1/2) + 6 
*a^4*c^3*d^5*e + 44*a^5*c^2*d^3*e^3 + 70*a^6*c*d*e^5 + 41*a*c^2*d^4*e^2*(- 
a^7*c)^(1/2) + 39*a^2*c*d^2*e^4*(-a^7*c)^(1/2))/(256*(a^11*e^8 + a^7*c^4*d 
^8 + 4*a^10*c*d^2*e^6 + 4*a^8*c^3*d^6*e^2 + 6*a^9*c^2*d^4*e^4)))^(1/2)*(65 
536*a^13*c^4*e^17 - 65536*a^6*c^11*d^14*e^3 - 327680*a^7*c^10*d^12*e^5 - 5 
89824*a^8*c^9*d^10*e^7 - 327680*a^9*c^8*d^8*e^9 + 327680*a^10*c^7*d^6*e^11 
 + 589824*a^11*c^6*d^4*e^13 + 327680*a^12*c^5*d^2*e^15))/(128*(a^8*e^8 + a 
^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(( 
9*c^3*d^6*(-a^7*c)^(1/2) - 25*a^3*e^6*(-a^7*c)^(1/2) + 6*a^4*c^3*d^5*e + 4 
4*a^5*c^2*d^3*e^3 + 70*a^6*c*d*e^5 + 41*a*c^2*d^4*e^2*(-a^7*c)^(1/2) + 39* 
a^2*c*d^2*e^4*(-a^7*c)^(1/2))/(256*(a^11*e^8 + a^7*c^4*d^8 + 4*a^10*c*d^2* 
e^6 + 4*a^8*c^3*d^6*e^2 + 6*a^9*c^2*d^4*e^4)))^(1/2) - (x*(1152*a^2*c^11*d 
^13*e^2 - 49024*a^8*c^5*d*e^14 + 7936*a^3*c^10*d^11*e^4 + 20352*a^4*c^9*d^ 
9*e^6 + 8704*a^5*c^8*d^7*e^8 - 66688*a^6*c^7*d^5*e^10 - 110848*a^7*c^6*d^3 
*e^12))/(128*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e...