\(\int \frac {1}{\sqrt {d+e x^2} (a-c x^4)} \, dx\) [345]

Optimal result

Integrand size = 22, antiderivative size = 135 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:

1/2*arctan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/ 
(c^(1/2)*d-a^(1/2)*e)^(1/2)+1/2*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1 
/4)/(e*x^2+d)^(1/2))/a^(3/4)/(c^(1/2)*d+a^(1/2)*e)^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx=2 e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-c d^3+3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}-3 c d \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

2*e^(3/2)*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 4*c* 
d*#1^3 + c*#1^4 & , (Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*# 
1)/(-(c*d^3) + 3*c*d^2*#1 - 8*a*e^2*#1 - 3*c*d*#1^2 + c*#1^3) & ]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1489, 27, 291, 218, 221}

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.

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

\(\Big \downarrow \) 1489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

\(\displaystyle \frac {\int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 \sqrt {a}}+\frac {\int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {\arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}\)

Input:

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

Output:

ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])]/(2*a^(3/ 
4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[a]*e]*x)/ 
(a^(1/4)*Sqrt[d + e*x^2])]/(2*a^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])
 

Defintions of rubi rules used

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 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 221

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

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(-\frac {\arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\)	\(102\)
default	\(\frac {-\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}}{2 \sqrt {a}}-\frac {-\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}}{2 \sqrt {a}}\)	\(780\)

Input:

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

Output:

-1/2/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctan((e*x^2+d)^(1/2)/x*a/((-a*e+(d 
^2*a*c)^(1/2))*a)^(1/2))+1/2/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctanh((e*x^ 
2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1225 vs. \(2 (95) = 190\).

Time = 0.28 (sec) , antiderivative size = 1225, normalized size of antiderivative = 9.07 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*sqrt(((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + 
a^5*e^4)) - e)/(a*c*d^2 - a^2*e^2))*log(((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a 
^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4))*x^2 + 2*e*x^2 + 2*(a*e*x + (a^2*c 
*d^2 - a^3*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4))*x)*s 
qrt(e*x^2 + d)*sqrt(((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c 
*d^2*e^2 + a^5*e^4)) - e)/(a*c*d^2 - a^2*e^2)) + d)/x^2) - 1/8*sqrt(((a*c* 
d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)) - e)/ 
(a*c*d^2 - a^2*e^2))*log(((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2* 
a^4*c*d^2*e^2 + a^5*e^4))*x^2 + 2*e*x^2 - 2*(a*e*x + (a^2*c*d^2 - a^3*e^2) 
*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4))*x)*sqrt(e*x^2 + d)* 
sqrt(((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5* 
e^4)) - e)/(a*c*d^2 - a^2*e^2)) + d)/x^2) - 1/8*sqrt(-((a*c*d^2 - a^2*e^2) 
*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)) + e)/(a*c*d^2 - a^2 
*e^2))*log(-((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 
 + a^5*e^4))*x^2 - 2*e*x^2 + 2*(a*e*x - (a^2*c*d^2 - a^3*e^2)*sqrt(c*d^2/( 
a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4))*x)*sqrt(e*x^2 + d)*sqrt(-((a*c*d 
^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)) + e)/( 
a*c*d^2 - a^2*e^2)) - d)/x^2) + 1/8*sqrt(-((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/ 
(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)) + e)/(a*c*d^2 - a^2*e^2))*log(- 
((a*c*d^2 - a^2*e^2)*sqrt(c*d^2/(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx=- \int \frac {1}{- a \sqrt {d + e x^{2}} + c x^{4} \sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

-Integral(1/(-a*sqrt(d + e*x**2) + c*x**4*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx=\int { -\frac {1}{{\left (c x^{4} - a\right )} \sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

-integrate(1/((c*x^4 - a)*sqrt(e*x^2 + d)), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a -\sqrt {e \,x^{2}+d}\, c \,x^{4}}d x \] Input:

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

Output:

int(1/(sqrt(d + e*x**2)*a - sqrt(d + e*x**2)*c*x**4),x)