\(\int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx\) [61]

Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}-\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \] Output:

1/2*arctan(b^(1/4)*x*(a^(1/2)+b^(1/2)*x^2)/a^(1/4)/(-b*x^4+a)^(1/2))/a^(1/ 
4)/b^(1/4)/c+1/2*arctanh(b^(1/4)*x*(a^(1/2)-b^(1/2)*x^2)/a^(1/4)/(-b*x^4+a 
)^(1/2))/a^(1/4)/b^(1/4)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (\arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a-b x^4}}\right )-i \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-b x^4}}{\sqrt [4]{a} \sqrt [4]{b} x}\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} c} \] Input:

Integrate[Sqrt[a - b*x^4]/(a*c + b*c*x^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {921}

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.

\(\displaystyle \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx\)

\(\Big \downarrow \) 921

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

Input:

Int[Sqrt[a - b*x^4]/(a*c + b*c*x^4),x]
 

Output:

ArcTan[(b^(1/4)*x*(Sqrt[a] + Sqrt[b]*x^2))/(a^(1/4)*Sqrt[a - b*x^4])]/(2*a 
^(1/4)*b^(1/4)*c) + ArcTanh[(b^(1/4)*x*(Sqrt[a] - Sqrt[b]*x^2))/(a^(1/4)*S 
qrt[a - b*x^4])]/(2*a^(1/4)*b^(1/4)*c)
 

Defintions of rubi rules used

rule 921

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = 
 Rt[(-a)*b, 4]}, Simp[(a/(2*c*q))*ArcTan[q*x*((a + q^2*x^2)/(a*Sqrt[a + b*x 
^4]))], x] + Simp[(a/(2*c*q))*ArcTanh[q*x*((a - q^2*x^2)/(a*Sqrt[a + b*x^4] 
))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]
 

Maple [A] (verified)

Time = 6.39 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {-b \,x^{4}+2 x^{2} \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}\, x +a}{-b \,x^{4}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}\, x +2 x^{2} \sqrt {a b}+a}\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}+1\right )}{8 \left (a b \right )^{\frac {1}{4}} c}\)	\(137\)
default	\(-\frac {\ln \left (\frac {\frac {-b \,x^{4}+a}{2 x^{2}}-\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}{\frac {-b \,x^{4}+a}{2 x^{2}}+\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}-1\right )}{8 c \left (a b \right )^{\frac {1}{4}}}\)	\(141\)
elliptic	\(-\frac {\ln \left (\frac {\frac {-b \,x^{4}+a}{2 x^{2}}-\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}{\frac {-b \,x^{4}+a}{2 x^{2}}+\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}-1\right )}{8 c \left (a b \right )^{\frac {1}{4}}}\)	\(141\)

Input:

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

Output:

-1/8/(a*b)^(1/4)*(ln((-b*x^4+2*x^2*(a*b)^(1/2)-2*(a*b)^(1/4)*(-b*x^4+a)^(1 
/2)*x+a)/(-b*x^4+2*(a*b)^(1/4)*(-b*x^4+a)^(1/2)*x+2*x^2*(a*b)^(1/2)+a))+2* 
arctan((-b*x^4+a)^(1/2)/x/(a*b)^(1/4)+1)-2*arctan(-(-b*x^4+a)^(1/2)/x/(a*b 
)^(1/4)+1))/c
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.16 \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} - \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} + 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) \] Input:

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

Output:

-1/4*(1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*log(-(4*(1/4)^(3/4)*a*b*c^3*x^3*(-1/ 
(a*b*c^4))^(3/4) + sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) - 2*(1/4)^(1/ 
4)*a*c*x*(-1/(a*b*c^4))^(1/4) + sqrt(-b*x^4 + a)*x^2)/(b*x^4 + a)) + 1/4*( 
1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*log((4*(1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^ 
4))^(3/4) - sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) - 2*(1/4)^(1/4)*a*c* 
x*(-1/(a*b*c^4))^(1/4) - sqrt(-b*x^4 + a)*x^2)/(b*x^4 + a)) - 1/4*I*(1/4)^ 
(1/4)*(-1/(a*b*c^4))^(1/4)*log((4*I*(1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^4)) 
^(3/4) + sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) + 2*I*(1/4)^(1/4)*a*c*x 
*(-1/(a*b*c^4))^(1/4) - sqrt(-b*x^4 + a)*x^2)/(b*x^4 + a)) + 1/4*I*(1/4)^( 
1/4)*(-1/(a*b*c^4))^(1/4)*log((-4*I*(1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^4)) 
^(3/4) + sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) - 2*I*(1/4)^(1/4)*a*c*x 
*(-1/(a*b*c^4))^(1/4) - sqrt(-b*x^4 + a)*x^2)/(b*x^4 + a))
 

Sympy [F]

\[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\int \frac {\sqrt {a - b x^{4}}}{a + b x^{4}}\, dx}{c} \] Input:

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

Output:

Integral(sqrt(a - b*x**4)/(a + b*x**4), x)/c
 

Maxima [F]

\[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{b c x^{4} + a c} \,d x } \] Input:

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

Output:

integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c), x)
 

Giac [F]

\[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{b c x^{4} + a c} \,d x } \] Input:

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

Output:

integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\int \frac {\sqrt {a-b\,x^4}}{b\,c\,x^4+a\,c} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\int \frac {\sqrt {-b \,x^{4}+a}}{b \,x^{4}+a}d x}{c} \] Input:

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

Output:

int(sqrt(a - b*x**4)/(a + b*x**4),x)/c