Integrand size = 13, antiderivative size = 60 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=-\frac {1}{2 a^2 x^2}-\frac {c x^2}{4 a^2 \left (a+c x^4\right )}-\frac {3 \sqrt {c} \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:
-1/2/a^2/x^2-1/4*c*x^2/a^2/(c*x^4+a)-3/4*c^(1/2)*arctan(c^(1/2)*x^2/a^(1/2 ))/a^(5/2)
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=\frac {-\frac {\sqrt {a} \left (2 a+3 c x^4\right )}{x^2 \left (a+c x^4\right )}+3 \sqrt {c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 \sqrt {c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{4 a^{5/2}} \] Input:
Integrate[1/(x^3*(a + c*x^4)^2),x]
Output:
(-((Sqrt[a]*(2*a + 3*c*x^4))/(x^2*(a + c*x^4))) + 3*Sqrt[c]*ArcTan[1 - (Sq rt[2]*c^(1/4)*x)/a^(1/4)] + 3*Sqrt[c]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/ 4)])/(4*a^(5/2))
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 253, 264, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (c x^4+a\right )^2}dx^2\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \int \frac {1}{x^4 \left (c x^4+a\right )}dx^2}{2 a}+\frac {1}{2 a x^2 \left (a+c x^4\right )}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (-\frac {c \int \frac {1}{c x^4+a}dx^2}{a}-\frac {1}{a x^2}\right )}{2 a}+\frac {1}{2 a x^2 \left (a+c x^4\right )}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x^2}\right )}{2 a}+\frac {1}{2 a x^2 \left (a+c x^4\right )}\right )\) |
Input:
Int[1/(x^3*(a + c*x^4)^2),x]
Output:
(1/(2*a*x^2*(a + c*x^4)) + (3*(-(1/(a*x^2)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2 )/Sqrt[a]])/a^(3/2)))/(2*a))/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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.47 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {1}{2 a^{2} x^{2}}-\frac {c \left (\frac {x^{2}}{2 c \,x^{4}+2 a}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{2 a^{2}}\) | \(49\) |
risch | \(\frac {-\frac {3 c \,x^{4}}{4 a^{2}}-\frac {1}{2 a}}{x^{2} \left (c \,x^{4}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{2}+c \right )}{\sum }\textit {\_R} \ln \left (\left (-5 a^{5} \textit {\_R}^{2}-4 c \right ) x^{2}-a^{3} \textit {\_R} \right )\right )}{8}\) | \(71\) |
Input:
int(1/x^3/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2/a^2/x^2-1/2/a^2*c*(1/2*x^2/(c*x^4+a)+3/2/(a*c)^(1/2)*arctan(c*x^2/(a* c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.47 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=\left [-\frac {6 \, c x^{4} - 3 \, {\left (c x^{6} + a x^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 4 \, a}{8 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )}}, -\frac {3 \, c x^{4} + 3 \, {\left (c x^{6} + a x^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right ) + 2 \, a}{4 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )}}\right ] \] Input:
integrate(1/x^3/(c*x^4+a)^2,x, algorithm="fricas")
Output:
[-1/8*(6*c*x^4 - 3*(c*x^6 + a*x^2)*sqrt(-c/a)*log((c*x^4 - 2*a*x^2*sqrt(-c /a) - a)/(c*x^4 + a)) + 4*a)/(a^2*c*x^6 + a^3*x^2), -1/4*(3*c*x^4 + 3*(c*x ^6 + a*x^2)*sqrt(c/a)*arctan(x^2*sqrt(c/a)) + 2*a)/(a^2*c*x^6 + a^3*x^2)]
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=\frac {3 \sqrt {- \frac {c}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {c}{a^{5}}}}{c} + x^{2} \right )}}{8} - \frac {3 \sqrt {- \frac {c}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {c}{a^{5}}}}{c} + x^{2} \right )}}{8} + \frac {- 2 a - 3 c x^{4}}{4 a^{3} x^{2} + 4 a^{2} c x^{6}} \] Input:
integrate(1/x**3/(c*x**4+a)**2,x)
Output:
3*sqrt(-c/a**5)*log(-a**3*sqrt(-c/a**5)/c + x**2)/8 - 3*sqrt(-c/a**5)*log( a**3*sqrt(-c/a**5)/c + x**2)/8 + (-2*a - 3*c*x**4)/(4*a**3*x**2 + 4*a**2*c *x**6)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=-\frac {3 \, c x^{4} + 2 \, a}{4 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )}} - \frac {3 \, c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} a^{2}} \] Input:
integrate(1/x^3/(c*x^4+a)^2,x, algorithm="maxima")
Output:
-1/4*(3*c*x^4 + 2*a)/(a^2*c*x^6 + a^3*x^2) - 3/4*c*arctan(c*x^2/sqrt(a*c)) /(sqrt(a*c)*a^2)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=-\frac {3 \, c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} a^{2}} - \frac {3 \, c x^{4} + 2 \, a}{4 \, {\left (c x^{6} + a x^{2}\right )} a^{2}} \] Input:
integrate(1/x^3/(c*x^4+a)^2,x, algorithm="giac")
Output:
-3/4*c*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a^2) - 1/4*(3*c*x^4 + 2*a)/((c*x ^6 + a*x^2)*a^2)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=-\frac {\frac {1}{2\,a}+\frac {3\,c\,x^4}{4\,a^2}}{c\,x^6+a\,x^2}-\frac {3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,a^{5/2}} \] Input:
int(1/(x^3*(a + c*x^4)^2),x)
Output:
- (1/(2*a) + (3*c*x^4)/(4*a^2))/(a*x^2 + c*x^6) - (3*c^(1/2)*atan((c^(1/2) *x^2)/a^(1/2)))/(4*a^(5/2))
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.97 \[ \int \frac {1}{x^3 \left (a+c x^4\right )^2} \, dx=\frac {3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,x^{6}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,x^{6}-2 a^{2}-3 a c \,x^{4}}{4 a^{3} x^{2} \left (c \,x^{4}+a \right )} \] Input:
int(1/x^3/(c*x^4+a)^2,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*x**2 + 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c*x**6 + 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*x**2 + 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*x**6 - 2*a**2 - 3*a*c*x**4)/(4*a**3*x* *2*(a + c*x**4))