Integrand size = 22, antiderivative size = 221 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\frac {c x \left (d-e x^2\right ) \sqrt {d+e x^2}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^4\right )}+\frac {\left (3 \sqrt {c} d-4 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (3 \sqrt {c} d+4 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \] Output:
1/4*c*x*(-e*x^2+d)*(e*x^2+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^4+a)+1/8*(3*c^(1 /2)*d-4*a^(1/2)*e)*arctan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^ (1/2))/a^(7/4)/(c^(1/2)*d-a^(1/2)*e)^(3/2)+1/8*(3*c^(1/2)*d+4*a^(1/2)*e)*a rctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(7/4)/(c^( 1/2)*d+a^(1/2)*e)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\frac {-2 c x \left (d-e x^2\right ) \sqrt {d+e x^2}+e^{3/2} \left (a-c x^4\right ) \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+10 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{8 a \left (-c d^2+a e^2\right ) \left (a-c x^4\right )} \] Input:
Integrate[1/(Sqrt[d + e*x^2]*(a - c*x^4)^2),x]
Output:
(-2*c*x*(d - e*x^2)*Sqrt[d + e*x^2] + e^(3/2)*(a - c*x^4)*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 & , (c*d^3 *Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 10*c*d^2*Log[d + 2* e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 16*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*S qrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/(8*a*(-(c*d^2) + a*e^2)*(a - c*x^4))
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.
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \sqrt {d+e x^2}}dx\) |
Input:
Int[1/(Sqrt[d + e*x^2]*(a - c*x^4)^2),x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
Time = 0.59 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.33
method | result | size |
pseudoelliptic | \(-\frac {\frac {\left (-c \,x^{4}+a \right ) \left (\left (4 a \,e^{2}-3 c \,d^{2}\right ) \sqrt {d^{2} a c}+d^{2} e a c \right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2}+\left (\frac {\left (\left (-4 a \,e^{2}+3 c \,d^{2}\right ) \sqrt {d^{2} a c}+d^{2} e a c \right ) \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2}+x \left (-e \,x^{2}+d \right ) \sqrt {e \,x^{2}+d}\, c \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}{4 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, a \left (a \,e^{2}-c \,d^{2}\right ) \left (-c \,x^{4}+a \right )}\) | \(294\) |
default | \(\text {Expression too large to display}\) | \(2200\) |
Input:
int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*(1/2*(-c*x^4+a)*((4*a*e^2-3*c*d^2)*(d^2*a*c)^(1/2)+d^2*e*a*c)*((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*((-4*a*e^2+3*c*d^2)*(d^2*a*c)^(1/2)+d^2*e*a*c)*(-c*x^4+a) *arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))+x*(-e*x^2+d) *(e*x^2+d)^(1/2)*c*(d^2*a*c)^(1/2)*((a*e+(d^2*a*c)^(1/2))*a)^(1/2))*((-a*e +(d^2*a*c)^(1/2))*a)^(1/2))/(d^2*a*c)^(1/2)/((-a*e+(d^2*a*c)^(1/2))*a)^(1/ 2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)/a/(a*e^2-c*d^2)/(-c*x^4+a)
Leaf count of result is larger than twice the leaf count of optimal. 4804 vs. \(2 (171) = 342\).
Time = 25.17 (sec) , antiderivative size = 4804, normalized size of antiderivative = 21.74 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x**2+d)**(1/2)/(-c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} - a\right )}^{2} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate(1/((c*x^4 - a)^2*sqrt(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^2\,\sqrt {e\,x^2+d}} \,d x \] Input:
int(1/((a - c*x^4)^2*(d + e*x^2)^(1/2)),x)
Output:
int(1/((a - c*x^4)^2*(d + e*x^2)^(1/2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a^{2}-2 \sqrt {e \,x^{2}+d}\, a c \,x^{4}+\sqrt {e \,x^{2}+d}\, c^{2} x^{8}}d x \] Input:
int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^2,x)
Output:
int(1/(sqrt(d + e*x**2)*a**2 - 2*sqrt(d + e*x**2)*a*c*x**4 + sqrt(d + e*x* *2)*c**2*x**8),x)