Integrand size = 38, antiderivative size = 329 \[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=-\frac {\left (c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \arctan \left (\frac {\sqrt {2} \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c}-\frac {\left (-c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c} \] Output:
-1/2*(c+(2*a^(1/2)*b^(1/2)+c^2)^(1/2))*(-a^(1/2)*b^(1/2)-c^2+c*(2*a^(1/2)* b^(1/2)+c^2)^(1/2))^(1/2)*arctan(2^(1/2)*(-a^(1/2)*b^(1/2)-c^2+c*(2*a^(1/2 )*b^(1/2)+c^2)^(1/2))^(1/2)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/ 2)/a^(1/2)/b^(1/2)/c-1/2*(-c+(2*a^(1/2)*b^(1/2)+c^2)^(1/2))*(a^(1/2)*b^(1/ 2)+c^2+c*(2*a^(1/2)*b^(1/2)+c^2)^(1/2))^(1/2)*arctanh(2^(1/2)*(a^(1/2)*b^( 1/2)+c^2+c*(2*a^(1/2)*b^(1/2)+c^2)^(1/2))^(1/2)*x/(b^(1/2)+a^(1/2)*x^2+(a* x^4+b)^(1/2)))*2^(1/2)/a^(1/2)/b^(1/2)/c
Time = 1.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.16 \[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=-\frac {\log \left (c x+\sqrt {b+a x^4}\right )}{2 c}+\frac {\log \left (c^2 x-c \sqrt {b+a x^4}\right )}{2 c} \] Input:
Integrate[(-b + a*x^4)/(Sqrt[b + a*x^4]*(b - c^2*x^2 + a*x^4)),x]
Output:
-1/2*Log[c*x + Sqrt[b + a*x^4]]/c + Log[c^2*x - c*Sqrt[b + a*x^4]]/(2*c)
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2537, 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.
\(\displaystyle \int \frac {a x^4-b}{\sqrt {a x^4+b} \left (a x^4+b-c^2 x^2\right )} \, dx\) |
\(\Big \downarrow \) 2537 |
\(\displaystyle -b \int \frac {1}{b-\frac {b c^2 x^2}{a x^4+b}}d\frac {x}{\sqrt {a x^4+b}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {c x}{\sqrt {a x^4+b}}\right )}{c}\) |
Input:
Int[(-b + a*x^4)/(Sqrt[b + a*x^4]*(b - c^2*x^2 + a*x^4)),x]
Output:
-(ArcTanh[(c*x)/Sqrt[b + a*x^4]]/c)
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]
Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Co eff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Simp[A Subst[Int[1/(d - (b*d - a*e )*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
Time = 0.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.07
method | result | size |
elliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{4}+b}}{x c}\right )}{c}\) | \(23\) |
pseudoelliptic | \(\frac {-\ln \left (\frac {c x +\sqrt {a \,x^{4}+b}}{x}\right )+\ln \left (\frac {-c x +\sqrt {a \,x^{4}+b}}{x}\right )}{2 c}\) | \(46\) |
default | \(\frac {-\ln \left (\frac {c x +\sqrt {a \,x^{4}+b}}{x}\right )+\ln \left (\frac {c x -\sqrt {a \,x^{4}+b}}{x}\right )}{2 c}\) | \(47\) |
Input:
int((a*x^4-b)/(a*x^4+b)^(1/2)/(a*x^4-c^2*x^2+b),x,method=_RETURNVERBOSE)
Output:
-1/c*arctanh((a*x^4+b)^(1/2)/x/c)
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.16 \[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=\frac {\log \left (\frac {a x^{4} + c^{2} x^{2} - 2 \, \sqrt {a x^{4} + b} c x + b}{a x^{4} - c^{2} x^{2} + b}\right )}{2 \, c} \] Input:
integrate((a*x^4-b)/(a*x^4+b)^(1/2)/(a*x^4-c^2*x^2+b),x, algorithm="fricas ")
Output:
1/2*log((a*x^4 + c^2*x^2 - 2*sqrt(a*x^4 + b)*c*x + b)/(a*x^4 - c^2*x^2 + b ))/c
\[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=\int \frac {a x^{4} - b}{\sqrt {a x^{4} + b} \left (a x^{4} + b - c^{2} x^{2}\right )}\, dx \] Input:
integrate((a*x**4-b)/(a*x**4+b)**(1/2)/(a*x**4-c**2*x**2+b),x)
Output:
Integral((a*x**4 - b)/(sqrt(a*x**4 + b)*(a*x**4 + b - c**2*x**2)), x)
\[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (a x^{4} - c^{2} x^{2} + b\right )} \sqrt {a x^{4} + b}} \,d x } \] Input:
integrate((a*x^4-b)/(a*x^4+b)^(1/2)/(a*x^4-c^2*x^2+b),x, algorithm="maxima ")
Output:
integrate((a*x^4 - b)/((a*x^4 - c^2*x^2 + b)*sqrt(a*x^4 + b)), x)
\[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (a x^{4} - c^{2} x^{2} + b\right )} \sqrt {a x^{4} + b}} \,d x } \] Input:
integrate((a*x^4-b)/(a*x^4+b)^(1/2)/(a*x^4-c^2*x^2+b),x, algorithm="giac")
Output:
integrate((a*x^4 - b)/((a*x^4 - c^2*x^2 + b)*sqrt(a*x^4 + b)), x)
Timed out. \[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=\int -\frac {b-a\,x^4}{\sqrt {a\,x^4+b}\,\left (-c^2\,x^2+a\,x^4+b\right )} \,d x \] Input:
int(-(b - a*x^4)/((b + a*x^4)^(1/2)*(b + a*x^4 - c^2*x^2)),x)
Output:
int(-(b - a*x^4)/((b + a*x^4)^(1/2)*(b + a*x^4 - c^2*x^2)), x)
\[ \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx=-\left (\int \frac {\sqrt {a \,x^{4}+b}}{a^{2} x^{8}-a \,c^{2} x^{6}+2 a b \,x^{4}-b \,c^{2} x^{2}+b^{2}}d x \right ) b +\left (\int \frac {\sqrt {a \,x^{4}+b}\, x^{4}}{a^{2} x^{8}-a \,c^{2} x^{6}+2 a b \,x^{4}-b \,c^{2} x^{2}+b^{2}}d x \right ) a \] Input:
int((a*x^4-b)/(a*x^4+b)^(1/2)/(a*x^4-c^2*x^2+b),x)
Output:
- int(sqrt(a*x**4 + b)/(a**2*x**8 + 2*a*b*x**4 - a*c**2*x**6 + b**2 - b*c **2*x**2),x)*b + int((sqrt(a*x**4 + b)*x**4)/(a**2*x**8 + 2*a*b*x**4 - a*c **2*x**6 + b**2 - b*c**2*x**2),x)*a