Integrand size = 11, antiderivative size = 68 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {x^2}{8 a \left (a+c x^4\right )^2}+\frac {3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}} \] Output:
1/8*x^2/a/(c*x^4+a)^2+3/16*x^2/a^2/(c*x^4+a)+3/16*arctan(c^(1/2)*x^2/a^(1/ 2))/a^(5/2)/c^(1/2)
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {1}{16} \left (\frac {5 a x^2+3 c x^6}{a^2 \left (a+c x^4\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{a^{5/2} \sqrt {c}}\right ) \] Input:
Integrate[x/(a + c*x^4)^3,x]
Output:
((5*a*x^2 + 3*c*x^6)/(a^2*(a + c*x^4)^2) + (3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a] ])/(a^(5/2)*Sqrt[c]))/16
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {807, 215, 215, 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.
\(\displaystyle \int \frac {x}{\left (a+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (c x^4+a\right )^3}dx^2\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \int \frac {1}{\left (c x^4+a\right )^2}dx^2}{4 a}+\frac {x^2}{4 a \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {\int \frac {1}{c x^4+a}dx^2}{2 a}+\frac {x^2}{2 a \left (a+c x^4\right )}\right )}{4 a}+\frac {x^2}{4 a \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {x^2}{2 a \left (a+c x^4\right )}\right )}{4 a}+\frac {x^2}{4 a \left (a+c x^4\right )^2}\right )\) |
Input:
Int[x/(a + c*x^4)^3,x]
Output:
(x^2/(4*a*(a + c*x^4)^2) + (3*(x^2/(2*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2 )/Sqrt[a]]/(2*a^(3/2)*Sqrt[c])))/(4*a))/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {x^{2}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (c \,x^{4}+a \right )}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 a \sqrt {a c}}}{a}\) | \(63\) |
risch | \(\frac {\frac {3 c \,x^{6}}{16 a^{2}}+\frac {5 x^{2}}{16 a}}{\left (c \,x^{4}+a \right )^{2}}-\frac {3 \ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, a^{2}}+\frac {3 \ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, a^{2}}\) | \(80\) |
Input:
int(x/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
1/8*x^2/a/(c*x^4+a)^2+3/8/a*(1/2*x^2/a/(c*x^4+a)+1/2/a/(a*c)^(1/2)*arctan( c*x^2/(a*c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.88 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\left [\frac {6 \, a c^{2} x^{6} + 10 \, a^{2} c x^{2} - 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{32 \, {\left (a^{3} c^{3} x^{8} + 2 \, a^{4} c^{2} x^{4} + a^{5} c\right )}}, \frac {3 \, a c^{2} x^{6} + 5 \, a^{2} c x^{2} - 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{16 \, {\left (a^{3} c^{3} x^{8} + 2 \, a^{4} c^{2} x^{4} + a^{5} c\right )}}\right ] \] Input:
integrate(x/(c*x^4+a)^3,x, algorithm="fricas")
Output:
[1/32*(6*a*c^2*x^6 + 10*a^2*c*x^2 - 3*(c^2*x^8 + 2*a*c*x^4 + a^2)*sqrt(-a* c)*log((c*x^4 - 2*sqrt(-a*c)*x^2 - a)/(c*x^4 + a)))/(a^3*c^3*x^8 + 2*a^4*c ^2*x^4 + a^5*c), 1/16*(3*a*c^2*x^6 + 5*a^2*c*x^2 - 3*(c^2*x^8 + 2*a*c*x^4 + a^2)*sqrt(a*c)*arctan(sqrt(a*c)/(c*x^2)))/(a^3*c^3*x^8 + 2*a^4*c^2*x^4 + a^5*c)]
Time = 0.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.62 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=- \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} c}} + x^{2} \right )}}{32} + \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} c}} + x^{2} \right )}}{32} + \frac {5 a x^{2} + 3 c x^{6}}{16 a^{4} + 32 a^{3} c x^{4} + 16 a^{2} c^{2} x^{8}} \] Input:
integrate(x/(c*x**4+a)**3,x)
Output:
-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x**2)/32 + 3*sqrt(-1/(a **5*c))*log(a**3*sqrt(-1/(a**5*c)) + x**2)/32 + (5*a*x**2 + 3*c*x**6)/(16* a**4 + 32*a**3*c*x**4 + 16*a**2*c**2*x**8)
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, c x^{6} + 5 \, a x^{2}}{16 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{2}} \] Input:
integrate(x/(c*x^4+a)^3,x, algorithm="maxima")
Output:
1/16*(3*c*x^6 + 5*a*x^2)/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 3/16*arctan(c *x^2/sqrt(a*c))/(sqrt(a*c)*a^2)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{6} + 5 \, a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \] Input:
integrate(x/(c*x^4+a)^3,x, algorithm="giac")
Output:
3/16*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/16*(3*c*x^6 + 5*a*x^2)/(( c*x^4 + a)^2*a^2)
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {5\,x^2}{16\,a}+\frac {3\,c\,x^6}{16\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,a^{5/2}\,\sqrt {c}} \] Input:
int(x/(a + c*x^4)^3,x)
Output:
((5*x^2)/(16*a) + (3*c*x^6)/(16*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) + (3*ata n((c^(1/2)*x^2)/a^(1/2)))/(16*a^(5/2)*c^(1/2))
Time = 0.22 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.01 \[ \int \frac {x}{\left (a+c x^4\right )^3} \, dx=\frac {-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}+5 a^{2} c \,x^{2}+3 a \,c^{2} x^{6}}{16 a^{3} c \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )} \] Input:
int(x/(c*x^4+a)^3,x)
Output:
( - 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**( 1/4)*a**(1/4)*sqrt(2)))*a**2 - 6*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**4 - 3*sqrt(c)*sq rt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sq rt(2)))*c**2*x**8 - 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2* sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2 - 6*sqrt(c)*sqrt(a)*atan((c** (1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x** 4 - 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**( 1/4)*a**(1/4)*sqrt(2)))*c**2*x**8 + 5*a**2*c*x**2 + 3*a*c**2*x**6)/(16*a** 3*c*(a**2 + 2*a*c*x**4 + c**2*x**8))