Integrand size = 33, antiderivative size = 141 \[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \]
arctan(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)-1/12*arctan(1/2*3^(1/4)*2^(3/4)* a^(1/4)*x/(a*x^4+b)^(1/4))*2^(1/4)*3^(3/4)/a^(1/4)+arctanh(a^(1/4)*x/(a*x^ 4+b)^(1/4))/a^(1/4)-1/12*arctanh(1/2*3^(1/4)*2^(3/4)*a^(1/4)*x/(a*x^4+b)^( 1/4))*2^(1/4)*3^(3/4)/a^(1/4)
Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {12 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\sqrt [4]{2} 3^{3/4} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+12 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\sqrt [4]{2} 3^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{12 \sqrt [4]{a}} \]
(12*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)] - 2^(1/4)*3^(3/4)*ArcTan[((3/2)^ (1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)] + 12*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^( 1/4)] - 2^(1/4)*3^(3/4)*ArcTanh[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)] )/(12*a^(1/4))
Time = 0.31 (sec) , antiderivative size = 160, 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.
| \(\displaystyle \int \frac {2 a x^4-3 b}{\left (a x^4-2 b\right ) \sqrt [4]{a x^4+b}} \, dx\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle 2 \int \frac {1}{\sqrt [4]{a x^4+b}}dx+b \int -\frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \int \frac {1}{\sqrt [4]{a x^4+b}}dx-b \int \frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle 2 \int \frac {1}{1-\frac {a x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}-b \int \frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}\right )-b \int \frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \int \frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \int \frac {1}{\left (2 b-a x^4\right ) \sqrt [4]{a x^4+b}}dx\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \int \frac {1}{2 b-\frac {3 a b x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} \sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {2} b}+\frac {\int \frac {1}{\frac {\sqrt {3} \sqrt {a} x^2}{\sqrt {a x^4+b}}+\sqrt {2}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {2} b}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} \sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {2} b}+\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a} b}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-b \left (\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a} b}\right )\) |
2*(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))) - b*(ArcTan[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*3^(1/4)*a^(1/4)*b) + ArcTanh[((3/2)^(1/4)*a^(1/4) *x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*3^(1/4)*a^(1/4)*b))
3.20.100.3.1 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 = 0.85 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(-\frac {\left (8 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) 6^{\frac {1}{4}}-2 \arctan \left (\frac {3^{\frac {3}{4}} 2^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{3 a^{\frac {1}{4}} x}\right ) \sqrt {2}-4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) 6^{\frac {1}{4}}+\ln \left (\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} a^{\frac {1}{4}} x +2 \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-2^{\frac {3}{4}} 3^{\frac {1}{4}} a^{\frac {1}{4}} x +2 \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) \sqrt {2}\right ) 6^{\frac {3}{4}}}{48 a^{\frac {1}{4}}}\) | \(160\) |
-1/48/a^(1/4)*(8*arctan(1/a^(1/4)/x*(a*x^4+b)^(1/4))*6^(1/4)-2*arctan(1/3* 3^(3/4)*2^(1/4)/a^(1/4)/x*(a*x^4+b)^(1/4))*2^(1/2)-4*ln((-a^(1/4)*x-(a*x^4 +b)^(1/4))/(a^(1/4)*x-(a*x^4+b)^(1/4)))*6^(1/4)+ln((2^(3/4)*3^(1/4)*a^(1/4 )*x+2*(a*x^4+b)^(1/4))/(-2^(3/4)*3^(1/4)*a^(1/4)*x+2*(a*x^4+b)^(1/4)))*2^( 1/2))*6^(3/4)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.68 \[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=-\frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (-\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {i \, \left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (\frac {12 i \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} - \frac {i \, \left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (\frac {-12 i \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \]
-1/4*(1/24)^(1/4)*log((12*(1/24)^(3/4)*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a ^(1/4) + 1/4*(1/24)^(1/4)*log(-(12*(1/24)^(3/4)*a^(1/4)*x - (a*x^4 + b)^(1 /4))/x)/a^(1/4) + 1/4*I*(1/24)^(1/4)*log((12*I*(1/24)^(3/4)*a^(1/4)*x + (a *x^4 + b)^(1/4))/x)/a^(1/4) - 1/4*I*(1/24)^(1/4)*log((-12*I*(1/24)^(3/4)*a ^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) + 1/2*log((a^(1/4)*x + (a*x^4 + b )^(1/4))/x)/a^(1/4) - 1/2*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/2*I*log((I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) + 1/2*I*log((-I*a ^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)
\[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int \frac {2 a x^{4} - 3 b}{\left (a x^{4} - 2 b\right ) \sqrt [4]{a x^{4} + b}}\, dx \]
\[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {2 \, a x^{4} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
\[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {2 \, a x^{4} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
Timed out. \[ \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int \frac {3\,b-2\,a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \]