Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
1/4*arctan(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(3/4)/a^(1/4)/b+1/4*arctan h(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(3/4)/a^(1/4)/b
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
(ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)] + ArcTanh[(2^(1/4)*a^(1/4) *x)/(-b + a*x^4)^(1/4)])/(2*2^(1/4)*a^(1/4)*b)
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {902, 756, 216, 219}
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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{a x^4-b} \left (a x^4+b\right )} \, dx\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \int \frac {1}{b-\frac {2 a b x^4}{a x^4-b}}d\frac {x}{\sqrt [4]{a x^4-b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 b}+\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 b}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 b}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}\) |
ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b) + Arc Tanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)
3.12.85.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Time = 1.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )\right )}{8 a^{\frac {1}{4}} b}\) | \(87\) |
1/8*2^(3/4)*(-2*arctan(1/2*2^(3/4)/a^(1/4)/x*(a*x^4-b)^(1/4))+ln((-2^(1/4) *a^(1/4)*x-(a*x^4-b)^(1/4))/(2^(1/4)*a^(1/4)*x-(a*x^4-b)^(1/4))))/a^(1/4)/ b
Result contains complex when optimal does not.
Time = 106.34 (sec) , antiderivative size = 525, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{4} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{4} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) \]
1/8*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(1/2*(4*(1/2)^(3/4)*sqrt(a*x^4 - b)*a *b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt( 1/(a*b^4)) + 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*(3*a*b*x^4 - b^2)*(1/(a*b ^4))^(1/4))/(a*x^4 + b)) - 1/8*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-1/2*(4*( 1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^ 4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/ 4)*(3*a*b*x^4 - b^2)*(1/(a*b^4))^(1/4))/(a*x^4 + b)) + 1/8*I*(1/2)^(1/4)*( 1/(a*b^4))^(1/4)*log(-1/2*(4*I*(1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a *b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2 *(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*(-3*I*a*b*x^4 + I*b^2)*(1/(a*b^4))^(1/4 ))/(a*x^4 + b)) - 1/8*I*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-1/2*(-4*I*(1/2) ^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*( 3*I*a*b*x^4 - I*b^2)*(1/(a*b^4))^(1/4))/(a*x^4 + b))
\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{4} + b\right )}\, dx \]
\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{1/4}} \,d x \]