Integrand size = 36, antiderivative size = 319 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3-2 \sqrt {2}} a}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3-2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{\sqrt {3-2 \sqrt {2}} a b}}{x}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3+2 \sqrt {2}} a}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3+2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{\sqrt {3+2 \sqrt {2}} a b}}{x}\right )}{a^5 b^5} \]
-1/4*x/a^4/b^4/(a^4*x^4-b^4)^(1/2)+(1/8-1/8*I)*arctan((1+I)*a*b*x/(I*b^2+a ^2*x^2+(a^4*x^4-b^4)^(1/2)))/a^5/b^5+(-1/16+1/16*I)*arctanh(((1/2+1/2*I)*b /(2^(1/2)-1)/a+(1/2-1/2*I)*a*x^2/(2^(1/2)-1)/b+(1/2-1/2*I)*(a^4*x^4-b^4)^( 1/2)/(2^(1/2)-1)/a/b)/x)/a^5/b^5+(1/16-1/16*I)*arctanh(((1/2+1/2*I)*b/(1+2 ^(1/2))/a+(1/2-1/2*I)*a*x^2/(1+2^(1/2))/b+(1/2-1/2*I)*(a^4*x^4-b^4)^(1/2)/ (1+2^(1/2))/a/b)/x)/a^5/b^5
Time = 0.83 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {-\frac {4 a b x}{\sqrt {-b^4+a^4 x^4}}+(2-2 i) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )+(1+i) \arctan \left (\frac {i b^4+(1-i) a b^3 x-(1+i) a^3 b x^3-i a^4 x^4+\left (b^2-(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{16 a^5 b^5} \]
((-4*a*b*x)/Sqrt[-b^4 + a^4*x^4] + (2 - 2*I)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])] + (1 + I)*ArcTan[(I*b^4 + (1 - I)*a*b^ 3*x - (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 + I)*a*b*x - I*a^2*x^2)*Sq rt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x ^4 + (b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(16*a^5*b^5 )
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1388, 971, 281, 921}
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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a^4 x^4-b^4} \left (a^8 x^8-b^8\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {x^4}{\left (a^4 x^4-b^4\right )^{3/2} \left (a^4 x^4+b^4\right )}dx\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle \frac {\int \frac {b^4-a^4 x^4}{\sqrt {a^4 x^4-b^4} \left (b^4+a^4 x^4\right )}dx}{4 a^4 b^4}-\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {\int \frac {\sqrt {a^4 x^4-b^4}}{b^4+a^4 x^4}dx}{4 a^4 b^4}-\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}\) |
| \(\Big \downarrow \) 921 |
| \(\displaystyle -\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}-\frac {-\frac {\arctan \left (\frac {a x \left (b^2-a^2 x^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{2 a b}-\frac {\text {arctanh}\left (\frac {a x \left (a^2 x^2+b^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{2 a b}}{4 a^4 b^4}\) |
-1/4*x/(a^4*b^4*Sqrt[-b^4 + a^4*x^4]) - (-1/2*ArcTan[(a*x*(b^2 - a^2*x^2)) /(b*Sqrt[-b^4 + a^4*x^4])]/(a*b) - ArcTanh[(a*x*(b^2 + a^2*x^2))/(b*Sqrt[- b^4 + a^4*x^4])]/(2*a*b))/(4*a^4*b^4)
3.29.99.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*b, 4]}, Simp[(a/(2*c*q))*ArcTan[q*x*((a + q^2*x^2)/(a*Sqrt[a + b*x ^4]))], x] + Simp[(a/(2*c*q))*ArcTanh[q*x*((a - q^2*x^2)/(a*Sqrt[a + b*x^4] ))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Time = 5.00 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.75
| method | result | size |
| elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(238\) |
| default | \(\frac {i \left (\left (-8 \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, a^{2} \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +i b \right ) \left (a x +b \right ) b^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right )}\) | \(508\) |
| pseudoelliptic | \(\frac {i \left (\left (-8 \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, a^{2} \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +i b \right ) \left (a x +b \right ) b^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right )}\) | \(508\) |
1/2*(-1/4/a^4/b^4/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x-1/32/a^4/b^4/(a^4*b^4)^(1/ 4)*2^(1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/ x+(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1 /2)/x+(a^4*b^4)^(1/2)))+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1 )+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)))*2^(1/2)
Time = 0.71 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.52 \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {4 \, \sqrt {a^{4} x^{4} - b^{4}} a b x + 2 \, {\left (a^{4} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\left (a^{4} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} + 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \]
-1/16*(4*sqrt(a^4*x^4 - b^4)*a*b*x + 2*(a^4*x^4 - b^4)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) - (a^4*x^4 - b^4)*log((a^4*x^4 + 2*a^2*b^2* x^2 - b^4 + 2*sqrt(a^4*x^4 - b^4)*a*b*x)/(a^4*x^4 + b^4)))/(a^9*b^5*x^4 - a^5*b^9)
\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}\, dx \]
Integral(x**4/(sqrt((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**2)*(a**4*x**4 + b**4)), x)
\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
\[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\int \frac {x^4}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \]