\(\int \frac {1}{(-2 b+a x^2) \sqrt [4]{-b+a x^2}} \, dx\) [1502]

Optimal result

Integrand size = 25, antiderivative size = 105 \[ \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \]

[Out]

1/4*arctan(1/a^(1/2)/x*2^(1/2)*b^(1/4)*(a*x^2-b)^(1/4))*2^(1/2)/a^(1/2)/b^(3/4)-1/4*arctanh(1/a^(1/2)/x*2^(1/2
)*b^(1/4)*(a*x^2-b)^(1/4))*2^(1/2)/a^(1/2)/b^(3/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {407} \[ \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \]

[In]

Int[1/((-2*b + a*x^2)*(-b + a*x^2)^(1/4)),x]

[Out]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \]

[In]

Integrate[1/((-2*b + a*x^2)*(-b + a*x^2)^(1/4)),x]

[Out]

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

Maple [F]

\[\int \frac {1}{\left (a \,x^{2}-2 b \right ) \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 68.46 (sec) , antiderivative size = 457, normalized size of antiderivative = 4.35 \[ \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, b}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} - i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, b}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} + i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, b}\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx=\int \frac {1}{\left (a x^{2} - 2 b\right ) \sqrt [4]{a x^{2} - b}}\, dx \]

[In]

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

[Out]

Integral(1/((a*x**2 - 2*b)*(a*x**2 - b)**(1/4)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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