Integrand size = 28, antiderivative size = 134 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d) (b e-a f)}-\frac {d^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d) (d e-c f)}+\frac {f^{3/2} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f) (d e-c f)} \] Output:
b^(3/2)*arctan(b^(1/2)*x/a^(1/2))/a^(1/2)/(-a*d+b*c)/(-a*f+b*e)-d^(3/2)*ar ctan(d^(1/2)*x/c^(1/2))/c^(1/2)/(-a*d+b*c)/(-c*f+d*e)+f^(3/2)*arctan(f^(1/ 2)*x/e^(1/2))/e^(1/2)/(-a*f+b*e)/(-c*f+d*e)
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (-b c+a d) (-b e+a f)}+\frac {d^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d) (-d e+c f)}+\frac {f^{3/2} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f) (d e-c f)} \] Input:
Integrate[1/((a + b*x^2)*(c + d*x^2)*(e + f*x^2)),x]
Output:
(b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)*(-(b*e) + a* f)) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)*(-(d*e) + c*f)) + (f^(3/2)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f)*(d*e - c*f))
Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {422, 303, 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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 422 |
\(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right ) \left (f x^2+e\right )}dx}{b c-a d}-\frac {d \int \frac {1}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{b c-a d}\) |
\(\Big \downarrow \) 303 |
\(\displaystyle \frac {b \left (\frac {b \int \frac {1}{b x^2+a}dx}{b e-a f}-\frac {f \int \frac {1}{f x^2+e}dx}{b e-a f}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {1}{f x^2+e}dx}{d e-c f}\right )}{b c-a d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {b \left (\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}\right )}{b c-a d}-\frac {d \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}\right )}{b c-a d}\) |
Input:
Int[1/((a + b*x^2)*(c + d*x^2)*(e + f*x^2)),x]
Output:
(b*((Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*e - a*f)) - (Sqrt[f] *ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f))))/(b*c - a*d) - (d*((S qrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) - (Sqrt[f]*ArcTa n[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f))))/(b*c - a*d)
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[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[b/(b *c - a*d) Int[1/(a + b*x^2), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x ^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d) Int[(c + d*x^2)^q*(e + f*x^2)^r, x], x] + Simp[b/(b*c - a*d) Int[(c + d*x^2)^(q + 1)*((e + f*x^2)^r/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
Time = 0.80 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a f -b e \right ) \left (a d -b c \right ) \sqrt {a b}}+\frac {f^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\left (c f -d e \right ) \left (a f -b e \right ) \sqrt {e f}}-\frac {d^{2} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {c d}}\) | \(117\) |
risch | \(\text {Expression too large to display}\) | \(4033\) |
Input:
int(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
b^2/(a*f-b*e)/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+f^2/(c*f-d*e)/ (a*f-b*e)/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2))-d^2/(a*d-b*c)/(c*f-d*e)/(c*d )^(1/2)*arctan(x*d/(c*d)^(1/2))
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)/(d*x**2+c)/(f*x**2+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c e - a b d e - a b c f + a^{2} d f\right )} \sqrt {a b}} - \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )} \sqrt {c d}} + \frac {f^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{{\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )} \sqrt {e f}} \] Input:
integrate(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="giac")
Output:
b^2*arctan(b*x/sqrt(a*b))/((b^2*c*e - a*b*d*e - a*b*c*f + a^2*d*f)*sqrt(a* b)) - d^2*arctan(d*x/sqrt(c*d))/((b*c*d*e - a*d^2*e - b*c^2*f + a*c*d*f)*s qrt(c*d)) + f^2*arctan(f*x/sqrt(e*f))/((b*d*e^2 - b*c*e*f - a*d*e*f + a*c* f^2)*sqrt(e*f))
Time = 6.37 (sec) , antiderivative size = 26278, normalized size of antiderivative = 196.10 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((a + b*x^2)*(c + d*x^2)*(e + f*x^2)),x)
Output:
(atan((((((-c*d^3)^(1/2)*((((x*(16*a^5*b^2*d^7*f^7 + 16*b^7*c^5*d^2*f^7 + 16*b^7*d^7*e^5*f^2 + 16*a^2*b^5*c^3*d^4*f^7 + 16*a^3*b^4*c^2*d^5*f^7 + 16* a^2*b^5*d^7*e^3*f^4 + 16*a^3*b^4*d^7*e^2*f^5 + 16*b^7*c^2*d^5*e^3*f^4 + 16 *b^7*c^3*d^4*e^2*f^5 - 32*a*b^6*c^4*d^3*f^7 - 32*a^4*b^3*c*d^6*f^7 - 32*a* b^6*d^7*e^4*f^3 - 32*a^4*b^3*d^7*e*f^6 - 32*b^7*c*d^6*e^4*f^3 - 32*b^7*c^4 *d^3*e*f^6 + 64*a*b^6*c*d^6*e^3*f^4 + 64*a*b^6*c^3*d^4*e*f^6 + 64*a^3*b^4* c*d^6*e*f^6 - 48*a*b^6*c^2*d^5*e^2*f^5 - 48*a^2*b^5*c*d^6*e^2*f^5 - 48*a^2 *b^5*c^2*d^5*e*f^6))/2 - ((-c*d^3)^(1/2)*((x*(-c*d^3)^(1/2)*(128*a^4*b^5*c ^6*d^3*f^9 - 32*a^3*b^6*c^7*d^2*f^9 - 192*a^5*b^4*c^5*d^4*f^9 + 128*a^6*b^ 3*c^4*d^5*f^9 - 32*a^7*b^2*c^3*d^6*f^9 - 32*a^3*b^6*d^9*e^7*f^2 + 128*a^4* b^5*d^9*e^6*f^3 - 192*a^5*b^4*d^9*e^5*f^4 + 128*a^6*b^3*d^9*e^4*f^5 - 32*a ^7*b^2*d^9*e^3*f^6 - 32*b^9*c^3*d^6*e^7*f^2 + 128*b^9*c^4*d^5*e^6*f^3 - 19 2*b^9*c^5*d^4*e^5*f^4 + 128*b^9*c^6*d^3*e^4*f^5 - 32*b^9*c^7*d^2*e^3*f^6 + 32*a*b^8*c^2*d^7*e^7*f^2 - 128*a*b^8*c^3*d^6*e^6*f^3 + 96*a*b^8*c^4*d^5*e ^5*f^4 + 96*a*b^8*c^5*d^4*e^4*f^5 - 128*a*b^8*c^6*d^3*e^3*f^6 + 32*a*b^8*c ^7*d^2*e^2*f^7 + 32*a^2*b^7*c*d^8*e^7*f^2 + 32*a^2*b^7*c^7*d^2*e*f^8 - 128 *a^3*b^6*c*d^8*e^6*f^3 - 128*a^3*b^6*c^6*d^3*e*f^8 + 96*a^4*b^5*c*d^8*e^5* f^4 + 96*a^4*b^5*c^5*d^4*e*f^8 + 96*a^5*b^4*c*d^8*e^4*f^5 + 96*a^5*b^4*c^4 *d^5*e*f^8 - 128*a^6*b^3*c*d^8*e^3*f^6 - 128*a^6*b^3*c^3*d^6*e*f^8 + 32*a^ 7*b^2*c*d^8*e^2*f^7 + 32*a^7*b^2*c^2*d^7*e*f^8 + 96*a^2*b^7*c^3*d^6*e^5...
Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,c^{2} e f -\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b c d \,e^{2}-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d e f +\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b d \,e^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a^{2} c d f -\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a b \,c^{2} f}{a c e \left (a^{2} c d \,f^{2}-a^{2} d^{2} e f -a b \,c^{2} f^{2}+a b \,d^{2} e^{2}+b^{2} c^{2} e f -b^{2} c d \,e^{2}\right )} \] Input:
int(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e),x)
Output:
(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*c**2*e*f - sqrt(b)*sqrt(a )*atan((b*x)/(sqrt(b)*sqrt(a)))*b*c*d*e**2 - sqrt(d)*sqrt(c)*atan((d*x)/(s qrt(d)*sqrt(c)))*a**2*d*e*f + sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)) )*a*b*d*e**2 + sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*f - sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*f)/(a*c*e*(a**2*c*d *f**2 - a**2*d**2*e*f - a*b*c**2*f**2 + a*b*d**2*e**2 + b**2*c**2*e*f - b* *2*c*d*e**2))