Integrand size = 33, antiderivative size = 133 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]
1/6*arctan(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)+1/6*arctan(2^(1/2)*a^(1/4)*x /(a*x^4+b)^(1/4))*2^(1/2)/a^(1/4)+1/6*arctanh(a^(1/4)*x/(a*x^4+b)^(1/4))/a ^(1/4)+1/6*arctanh(2^(1/2)*a^(1/4)*x/(a*x^4+b)^(1/4))*2^(1/2)/a^(1/4)
Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}} \]
(ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*a^(1/4)*x )/(b + a*x^4)^(1/4)] + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)] + Sqrt[2]*Ar cTanh[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)])/(6*a^(1/4))
Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1026, 25, 770, 756, 216, 219, 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.
| \(\displaystyle \int \frac {a x^4-b}{\sqrt [4]{a x^4+b} \left (3 a x^4-b\right )} \, dx\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\sqrt [4]{a x^4+b}}dx-\frac {2}{3} b \int -\frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\sqrt [4]{a x^4+b}}dx+\frac {2}{3} b \int \frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {2}{3} b \int \frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx+\frac {1}{3} \int \frac {1}{1-\frac {a x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2}{3} b \int \frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx+\frac {1}{3} \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )+\frac {2}{3} b \int \frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{3} b \int \frac {1}{\left (b-3 a x^4\right ) \sqrt [4]{a x^4+b}}dx+\frac {1}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {2}{3} b \int \frac {1}{b-\frac {4 a b x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2}{3} b \left (\frac {\int \frac {1}{1-\frac {2 \sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 b}+\frac {\int \frac {1}{\frac {2 \sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 b}\right )+\frac {1}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2}{3} b \left (\frac {\int \frac {1}{1-\frac {2 \sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 b}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b}\right )+\frac {1}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )+\frac {2}{3} b \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b}\right )\) |
(ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/( b + a*x^4)^(1/4)]/(2*a^(1/4)))/3 + (2*b*(ArcTan[(Sqrt[2]*a^(1/4)*x)/(b + a *x^4)^(1/4)]/(2*Sqrt[2]*a^(1/4)*b) + ArcTanh[(Sqrt[2]*a^(1/4)*x)/(b + a*x^ 4)^(1/4)]/(2*Sqrt[2]*a^(1/4)*b)))/3
3.20.14.3.1 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_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
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]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Time = 0.58 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) \sqrt {2}+\ln \left (\frac {x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{12 a^{\frac {1}{4}}}\) | \(136\) |
1/12*(-2*arctan(1/2*2^(1/2)/a^(1/4)/x*(a*x^4+b)^(1/4))*2^(1/2)+ln((x*a^(1/ 4)*2^(1/2)+(a*x^4+b)^(1/4))/(-x*a^(1/4)*2^(1/2)+(a*x^4+b)^(1/4)))*2^(1/2)- 2*arctan(1/a^(1/4)/x*(a*x^4+b)^(1/4))+ln((-a^(1/4)*x-(a*x^4+b)^(1/4))/(a^( 1/4)*x-(a*x^4+b)^(1/4))))/a^(1/4)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.78 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} \]
1/6*(1/4)^(1/4)*log((4*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/ 4) - 1/6*(1/4)^(1/4)*log(-(4*(1/4)^(3/4)*a^(1/4)*x - (a*x^4 + b)^(1/4))/x) /a^(1/4) - 1/6*I*(1/4)^(1/4)*log((4*I*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 + b)^ (1/4))/x)/a^(1/4) + 1/6*I*(1/4)^(1/4)*log((-4*I*(1/4)^(3/4)*a^(1/4)*x + (a *x^4 + b)^(1/4))/x)/a^(1/4) + 1/12*log((a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/ a^(1/4) - 1/12*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/12*I*lo g((I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) + 1/12*I*log((-I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)
\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {a x^{4} - b}{\sqrt [4]{a x^{4} + b} \left (3 a x^{4} - b\right )}\, dx \]
\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {b-a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-3\,a\,x^4\right )} \,d x \]