Integrand size = 26, antiderivative size = 141 \[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{7/4}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{7/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}} \]
-1/2*arctan(a^(1/4)*x/(a*x^4-b)^(1/4))/a^(7/4)+1/4*arctan(2^(1/4)*a^(1/4)* x/(a*x^4-b)^(1/4))*2^(1/4)/a^(7/4)+1/2*arctanh(a^(1/4)*x/(a*x^4-b)^(1/4))/ a^(7/4)-1/4*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(1/4)/a^(7/4)
Time = 0.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=\frac {-2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 a^{7/4}} \]
(-2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + 2^(1/4)*ArcTan[(2^(1/4)*a^(1/ 4)*x)/(-b + a*x^4)^(1/4)] + 2*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] - 2^ (1/4)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(4*a^(7/4))
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {996, 27, 981, 827, 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.
| \(\displaystyle \int \frac {x^6}{\left (a x^4-b\right )^{3/4} \left (a x^4+b\right )} \, dx\) |
| \(\Big \downarrow \) 996 |
| \(\displaystyle -b \int \frac {x^6}{b \left (a x^4-b\right )^{3/2} \left (1-\frac {2 a x^4}{a x^4-b}\right ) \left (1-\frac {a x^4}{a x^4-b}\right )}d\frac {x}{\sqrt [4]{a x^4-b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {x^6}{\left (a x^4-b\right )^{3/2} \left (1-\frac {2 a x^4}{a x^4-b}\right ) \left (1-\frac {a x^4}{a x^4-b}\right )}d\frac {x}{\sqrt [4]{a x^4-b}}\) |
| \(\Big \downarrow \) 981 |
| \(\displaystyle \frac {\int \frac {x^2}{\sqrt {a x^4-b} \left (1-\frac {a x^4}{a x^4-b}\right )}d\frac {x}{\sqrt [4]{a x^4-b}}}{a}-\frac {\int \frac {x^2}{\sqrt {a x^4-b} \left (1-\frac {2 a x^4}{a x^4-b}\right )}d\frac {x}{\sqrt [4]{a x^4-b}}}{a}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\frac {\int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a}}}{a}-\frac {\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {2} \sqrt {a}}}{a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{3/4}}}{a}-\frac {\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {2} \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{3/4}}}{a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{3/4}}}{a}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{3/4}}}{a}\) |
(-1/2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x) /(-b + a*x^4)^(1/4)]/(2*a^(3/4)))/a - (-1/2*ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2^(3/4)*a^(3/4)) + ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x ^4)^(1/4)]/(2*2^(3/4)*a^(3/4)))/a
3.20.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Simp[(-a)*(e^n/(b*c - a*d)) Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Simp[c*(e^n/(b*c - a*d)) Int[(e*x)^(m - n)/(c + d*x^n), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.) , x_Symbol] :> With[{k = Denominator[p]}, Simp[k*(a^(p + (m + 1)/n)/n) Su bst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p + q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && RationalQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]
Time = 0.76 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {-2 \,2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )-2^{\frac {1}{4}} \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )+2 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )+4 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{8 a^{\frac {7}{4}}}\) | \(151\) |
1/8*(-2*2^(1/4)*arctan(1/2*2^(3/4)/a^(1/4)/x*(a*x^4-b)^(1/4))-2^(1/4)*ln(( -x*2^(1/4)*a^(1/4)-(a*x^4-b)^(1/4))/(x*2^(1/4)*a^(1/4)-(a*x^4-b)^(1/4)))+2 *ln((a^(1/4)*x+(a*x^4-b)^(1/4))/(-a^(1/4)*x+(a*x^4-b)^(1/4)))+4*arctan(1/a ^(1/4)/x*(a*x^4-b)^(1/4)))/a^(7/4)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.19 \[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} i \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {i \, a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} i \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {-i \, a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/4*(1/8)^(1/4)*(a^(-7))^(1/4)*log((2*(1/8)^(1/4)*a^2*(a^(-7))^(1/4)*x + (a*x^4 - b)^(1/4))/x) + 1/4*(1/8)^(1/4)*(a^(-7))^(1/4)*log(-(2*(1/8)^(1/4) *a^2*(a^(-7))^(1/4)*x - (a*x^4 - b)^(1/4))/x) - 1/4*I*(1/8)^(1/4)*(a^(-7)) ^(1/4)*log((2*I*(1/8)^(1/4)*a^2*(a^(-7))^(1/4)*x + (a*x^4 - b)^(1/4))/x) + 1/4*I*(1/8)^(1/4)*(a^(-7))^(1/4)*log((-2*I*(1/8)^(1/4)*a^2*(a^(-7))^(1/4) *x + (a*x^4 - b)^(1/4))/x) + 1/4*(a^(-7))^(1/4)*log((a^2*(a^(-7))^(1/4)*x + (a*x^4 - b)^(1/4))/x) - 1/4*(a^(-7))^(1/4)*log(-(a^2*(a^(-7))^(1/4)*x - (a*x^4 - b)^(1/4))/x) + 1/4*I*(a^(-7))^(1/4)*log((I*a^2*(a^(-7))^(1/4)*x + (a*x^4 - b)^(1/4))/x) - 1/4*I*(a^(-7))^(1/4)*log((-I*a^2*(a^(-7))^(1/4)*x + (a*x^4 - b)^(1/4))/x)
\[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=\int \frac {x^{6}}{\left (a x^{4} - b\right )^{\frac {3}{4}} \left (a x^{4} + b\right )}\, dx \]
\[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=\int { \frac {x^{6}}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}} \,d x } \]
\[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=\int { \frac {x^{6}}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx=\int \frac {x^6}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{3/4}} \,d x \]