Integrand size = 27, antiderivative size = 95 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {(A b-a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (b c-a d)}+\frac {(B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)} \] Output:
1/2*(A*b-B*a)*arctan(b^(1/2)*x^2/a^(1/2))/a^(1/2)/b^(1/2)/(-a*d+b*c)+1/2*( -A*d+B*c)*arctan(d^(1/2)*x^2/c^(1/2))/c^(1/2)/d^(1/2)/(-a*d+b*c)
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {(-A b+a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (-b c+a d)}+\frac {(B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)} \] Input:
Integrate[(x*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
Output:
((-(A*b) + a*B)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*(-(b*c) + a*d)) + ((B*c - A*d)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*Sqrt[c]*Sqrt[d]*( b*c - a*d))
Time = 0.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1045, 397, 218}
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.
\(\displaystyle \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\) |
\(\Big \downarrow \) 1045 |
\(\displaystyle \frac {1}{2} \int \frac {B x^4+A}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {1}{2} \left (\frac {(A b-a B) \int \frac {1}{b x^4+a}dx^2}{b c-a d}+\frac {(B c-A d) \int \frac {1}{d x^4+c}dx^2}{b c-a d}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {(A b-a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} (b c-a d)}+\frac {(B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (b c-a d)}\right )\) |
Input:
Int[(x*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
Output:
(((A*b - a*B)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(b*c - a*d)) + ((B*c - A*d)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]*(b*c - a*d )))/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. )*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Si mp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q*(e + f*x^(n/k))^r, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\left (A b -B a \right ) \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -c b \right ) \sqrt {a b}}+\frac {\left (A d -B c \right ) \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -c b \right ) \sqrt {c d}}\) | \(74\) |
risch | \(-\frac {\ln \left (-\left (-c d \right )^{\frac {3}{2}} x^{2}-c^{2} d \right ) A d}{4 \sqrt {-c d}\, \left (a d -c b \right )}+\frac {\ln \left (-\left (-c d \right )^{\frac {3}{2}} x^{2}-c^{2} d \right ) B c}{4 \sqrt {-c d}\, \left (a d -c b \right )}+\frac {\ln \left (-\left (-c d \right )^{\frac {3}{2}} x^{2}+c^{2} d \right ) A d}{4 \sqrt {-c d}\, \left (a d -c b \right )}-\frac {\ln \left (-\left (-c d \right )^{\frac {3}{2}} x^{2}+c^{2} d \right ) B c}{4 \sqrt {-c d}\, \left (a d -c b \right )}-\frac {\ln \left (-\left (-a b \right )^{\frac {3}{2}} x^{2}+a^{2} b \right ) A b}{4 \sqrt {-a b}\, \left (a d -c b \right )}+\frac {\ln \left (-\left (-a b \right )^{\frac {3}{2}} x^{2}+a^{2} b \right ) B a}{4 \sqrt {-a b}\, \left (a d -c b \right )}+\frac {\ln \left (-\left (-a b \right )^{\frac {3}{2}} x^{2}-a^{2} b \right ) A b}{4 \sqrt {-a b}\, \left (a d -c b \right )}-\frac {\ln \left (-\left (-a b \right )^{\frac {3}{2}} x^{2}-a^{2} b \right ) B a}{4 \sqrt {-a b}\, \left (a d -c b \right )}\) | \(310\) |
Input:
int(x*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
Output:
-1/2*(A*b-B*a)/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))+1/2*(A*d-B* c)/(a*d-b*c)/(c*d)^(1/2)*arctan(d*x^2/(c*d)^(1/2))
Time = 1.85 (sec) , antiderivative size = 421, normalized size of antiderivative = 4.43 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [-\frac {{\left (B a - A b\right )} \sqrt {-a b} c d \log \left (\frac {b x^{4} + 2 \, \sqrt {-a b} x^{2} - a}{b x^{4} + a}\right ) + {\left (B a b c - A a b d\right )} \sqrt {-c d} \log \left (\frac {d x^{4} - 2 \, \sqrt {-c d} x^{2} - c}{d x^{4} + c}\right )}{4 \, {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )}}, -\frac {{\left (B a - A b\right )} \sqrt {-a b} c d \log \left (\frac {b x^{4} + 2 \, \sqrt {-a b} x^{2} - a}{b x^{4} + a}\right ) + 2 \, {\left (B a b c - A a b d\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d}}{d x^{2}}\right )}{4 \, {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )}}, \frac {2 \, {\left (B a - A b\right )} \sqrt {a b} c d \arctan \left (\frac {\sqrt {a b}}{b x^{2}}\right ) - {\left (B a b c - A a b d\right )} \sqrt {-c d} \log \left (\frac {d x^{4} - 2 \, \sqrt {-c d} x^{2} - c}{d x^{4} + c}\right )}{4 \, {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )}}, \frac {{\left (B a - A b\right )} \sqrt {a b} c d \arctan \left (\frac {\sqrt {a b}}{b x^{2}}\right ) - {\left (B a b c - A a b d\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d}}{d x^{2}}\right )}{2 \, {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )}}\right ] \] Input:
integrate(x*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
Output:
[-1/4*((B*a - A*b)*sqrt(-a*b)*c*d*log((b*x^4 + 2*sqrt(-a*b)*x^2 - a)/(b*x^ 4 + a)) + (B*a*b*c - A*a*b*d)*sqrt(-c*d)*log((d*x^4 - 2*sqrt(-c*d)*x^2 - c )/(d*x^4 + c)))/(a*b^2*c^2*d - a^2*b*c*d^2), -1/4*((B*a - A*b)*sqrt(-a*b)* c*d*log((b*x^4 + 2*sqrt(-a*b)*x^2 - a)/(b*x^4 + a)) + 2*(B*a*b*c - A*a*b*d )*sqrt(c*d)*arctan(sqrt(c*d)/(d*x^2)))/(a*b^2*c^2*d - a^2*b*c*d^2), 1/4*(2 *(B*a - A*b)*sqrt(a*b)*c*d*arctan(sqrt(a*b)/(b*x^2)) - (B*a*b*c - A*a*b*d) *sqrt(-c*d)*log((d*x^4 - 2*sqrt(-c*d)*x^2 - c)/(d*x^4 + c)))/(a*b^2*c^2*d - a^2*b*c*d^2), 1/2*((B*a - A*b)*sqrt(a*b)*c*d*arctan(sqrt(a*b)/(b*x^2)) - (B*a*b*c - A*a*b*d)*sqrt(c*d)*arctan(sqrt(c*d)/(d*x^2)))/(a*b^2*c^2*d - a ^2*b*c*d^2)]
Timed out. \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(x*(B*x**4+A)/(b*x**4+a)/(d*x**4+c),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a - A b\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {{\left (B c - A d\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \] Input:
integrate(x*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
Output:
-1/2*(B*a - A*b)*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + 1/2*(B* c - A*d)*arctan(d*x^2/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a - A b\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {{\left (B c - A d\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \] Input:
integrate(x*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
Output:
-1/2*(B*a - A*b)*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + 1/2*(B* c - A*d)*arctan(d*x^2/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 7.93 (sec) , antiderivative size = 1201, normalized size of antiderivative = 12.64 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx =\text {Too large to display} \] Input:
int((x*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x)
Output:
(B*a*c*d*atan((A^2*a^2*b^4*d^2*x^2*1i + B^2*a^2*b^4*c^2*x^2*1i - B^2*a^3*b ^3*c*d*x^2*1i - A^2*a*b^5*c*d*x^2*1i)/(B^2*a^3*d^2*(-a*b)^(3/2) + A^2*b^3* c*d*(-a*b)^(3/2) + A^2*a*b^2*d^2*(-a*b)^(3/2) + B^2*a^4*b*d^2*(-a*b)^(1/2) + 2*A^2*a^2*b^3*d^2*(-a*b)^(1/2) + B^2*a^2*b^3*c^2*(-a*b)^(1/2) - 2*A*B*a ^2*b*d^2*(-a*b)^(3/2) + B^2*a^2*b*c*d*(-a*b)^(3/2) - 2*A*B*a^3*b^2*d^2*(-a *b)^(1/2) - 2*A*B*a*b^2*c*d*(-a*b)^(3/2) - 2*A*B*a^2*b^3*c*d*(-a*b)^(1/2)) )*(-a*b)^(1/2)*1i)/(2*a*b^2*c^2*d - 2*a^2*b*c*d^2) - (A*b*c*d*atan((A^2*a^ 2*b^4*d^2*x^2*1i + B^2*a^2*b^4*c^2*x^2*1i - B^2*a^3*b^3*c*d*x^2*1i - A^2*a *b^5*c*d*x^2*1i)/(B^2*a^3*d^2*(-a*b)^(3/2) + A^2*b^3*c*d*(-a*b)^(3/2) + A^ 2*a*b^2*d^2*(-a*b)^(3/2) + B^2*a^4*b*d^2*(-a*b)^(1/2) + 2*A^2*a^2*b^3*d^2* (-a*b)^(1/2) + B^2*a^2*b^3*c^2*(-a*b)^(1/2) - 2*A*B*a^2*b*d^2*(-a*b)^(3/2) + B^2*a^2*b*c*d*(-a*b)^(3/2) - 2*A*B*a^3*b^2*d^2*(-a*b)^(1/2) - 2*A*B*a*b ^2*c*d*(-a*b)^(3/2) - 2*A*B*a^2*b^3*c*d*(-a*b)^(1/2)))*(-a*b)^(1/2)*1i)/(2 *a*b^2*c^2*d - 2*a^2*b*c*d^2) + (A*a*b*d*atan((A^2*b^2*c^2*d^4*x^2*1i + B^ 2*a^2*c^2*d^4*x^2*1i - B^2*a*b*c^3*d^3*x^2*1i - A^2*a*b*c*d^5*x^2*1i)/(B^2 *b^2*c^3*(-c*d)^(3/2) + A^2*a*b*d^3*(-c*d)^(3/2) + A^2*b^2*c*d^2*(-c*d)^(3 /2) + B^2*b^2*c^4*d*(-c*d)^(1/2) + 2*A^2*b^2*c^2*d^3*(-c*d)^(1/2) + B^2*a^ 2*c^2*d^3*(-c*d)^(1/2) - 2*A*B*b^2*c^2*d*(-c*d)^(3/2) + B^2*a*b*c^2*d*(-c* d)^(3/2) - 2*A*B*b^2*c^3*d^2*(-c*d)^(1/2) - 2*A*B*a*b*c*d^2*(-c*d)^(3/2) - 2*A*B*a*b*c^2*d^3*(-c*d)^(1/2)))*(-c*d)^(1/2)*1i)/(2*a*b^2*c^2*d - 2*a...
Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {\sqrt {d}\, \sqrt {c}\, \left (\mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )+\mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )\right )}{2 c d} \] Input:
int(x*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x)
Output:
( - sqrt(d)*sqrt(c)*(atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1 /4)*c**(1/4)*sqrt(2))) + atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(d)*x)/(d **(1/4)*c**(1/4)*sqrt(2)))))/(2*c*d)