\(\int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx\) [192]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.02 (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 = {414} \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}-\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]

[In]

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

[Out]

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)*Sqrt[a - b*x^4])]/(2*a^(1/4)*b^(1/4)*c)

Rule 414

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*b, 4]}, Simp[(a/(2*c*q))*A
rcTan[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\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 {\tanh ^{-1}\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (\arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a-b x^4}}\right )-i \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-b x^4}}{\sqrt [4]{a} \sqrt [4]{b} x}\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 6.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.22

method	result	size
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	\(-\frac {\ln \left (\frac {\frac {-b \,x^{4}+a}{2 x^{2}}-\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}{\frac {-b \,x^{4}+a}{2 x^{2}}+\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}-1\right )}{8 c \left (a b \right )^{\frac {1}{4}}}\)	\(141\)
pseudoelliptic	\(-\frac {\ln \left (\frac {-b \,x^{4}+2 x^{2} \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}\, x +a}{-b \,x^{4}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}\, x +2 x^{2} \sqrt {a b}+a}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {-x \left (a b \right )^{\frac {1}{4}}+\sqrt {-b \,x^{4}+a}}{x \left (a b \right )^{\frac {1}{4}}}\right )}{8 \left (a b \right )^{\frac {1}{4}} c}\)	\(149\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.16 \[ \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} - \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} + 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) \]

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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