Integrand size = 13, antiderivative size = 71 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=-\frac {a x^2}{c^3}+\frac {x^6}{6 c^2}-\frac {a^2 x^2}{4 c^3 \left (a+c x^4\right )}+\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{7/2}} \] Output:
-a*x^2/c^3+1/6*x^6/c^2-1/4*a^2*x^2/c^3/(c*x^4+a)+5/4*a^(3/2)*arctan(c^(1/2 )*x^2/a^(1/2))/c^(7/2)
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=\frac {x^2 \left (-12 a+2 c x^4-\frac {3 a^2}{a+c x^4}\right )}{12 c^3}+\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{7/2}} \] Input:
Integrate[x^13/(a + c*x^4)^2,x]
Output:
(x^2*(-12*a + 2*c*x^4 - (3*a^2)/(a + c*x^4)))/(12*c^3) + (5*a^(3/2)*ArcTan [(Sqrt[c]*x^2)/Sqrt[a]])/(4*c^(7/2))
Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 252, 254, 2009}
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.
\(\displaystyle \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (c x^4+a\right )^2}dx^2\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {1}{2} \left (\frac {5 \int \frac {x^8}{c x^4+a}dx^2}{2 c}-\frac {x^{10}}{2 c \left (a+c x^4\right )}\right )\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {1}{2} \left (\frac {5 \int \left (\frac {x^4}{c}+\frac {a^2}{c^2 \left (c x^4+a\right )}-\frac {a}{c^2}\right )dx^2}{2 c}-\frac {x^{10}}{2 c \left (a+c x^4\right )}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {5 \left (\frac {a^{3/2} \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{c^{5/2}}-\frac {a x^2}{c^2}+\frac {x^6}{3 c}\right )}{2 c}-\frac {x^{10}}{2 c \left (a+c x^4\right )}\right )\) |
Input:
Int[x^13/(a + c*x^4)^2,x]
Output:
(-1/2*x^10/(c*(a + c*x^4)) + (5*(-((a*x^2)/c^2) + x^6/(3*c) + (a^(3/2)*Arc Tan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(5/2)))/(2*c))/2
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
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 = 0.46 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {-\frac {1}{6} c \,x^{6}+a \,x^{2}}{c^{3}}+\frac {a^{2} \left (-\frac {x^{2}}{2 \left (c \,x^{4}+a \right )}+\frac {5 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{2 c^{3}}\) | \(60\) |
risch | \(\frac {x^{6}}{6 c^{2}}-\frac {a \,x^{2}}{c^{3}}-\frac {a^{2} x^{2}}{4 c^{3} \left (c \,x^{4}+a \right )}+\frac {5 \sqrt {-a c}\, a \ln \left (c \,x^{2}+\sqrt {-a c}\right )}{8 c^{4}}-\frac {5 \sqrt {-a c}\, a \ln \left (c \,x^{2}-\sqrt {-a c}\right )}{8 c^{4}}\) | \(91\) |
Input:
int(x^13/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/c^3*(-1/6*c*x^6+a*x^2)+1/2/c^3*a^2*(-1/2*x^2/(c*x^4+a)+5/2/(a*c)^(1/2)* arctan(c*x^2/(a*c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.42 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=\left [\frac {4 \, c^{2} x^{10} - 20 \, a c x^{6} - 30 \, a^{2} x^{2} + 15 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right )}{24 \, {\left (c^{4} x^{4} + a c^{3}\right )}}, \frac {2 \, c^{2} x^{10} - 10 \, a c x^{6} - 15 \, a^{2} x^{2} + 15 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right )}{12 \, {\left (c^{4} x^{4} + a c^{3}\right )}}\right ] \] Input:
integrate(x^13/(c*x^4+a)^2,x, algorithm="fricas")
Output:
[1/24*(4*c^2*x^10 - 20*a*c*x^6 - 30*a^2*x^2 + 15*(a*c*x^4 + a^2)*sqrt(-a/c )*log((c*x^4 + 2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)))/(c^4*x^4 + a*c^3), 1/ 12*(2*c^2*x^10 - 10*a*c*x^6 - 15*a^2*x^2 + 15*(a*c*x^4 + a^2)*sqrt(a/c)*ar ctan(c*x^2*sqrt(a/c)/a))/(c^4*x^4 + a*c^3)]
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.58 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=- \frac {a^{2} x^{2}}{4 a c^{3} + 4 c^{4} x^{4}} - \frac {a x^{2}}{c^{3}} - \frac {5 \sqrt {- \frac {a^{3}}{c^{7}}} \log {\left (x^{2} - \frac {c^{3} \sqrt {- \frac {a^{3}}{c^{7}}}}{a} \right )}}{8} + \frac {5 \sqrt {- \frac {a^{3}}{c^{7}}} \log {\left (x^{2} + \frac {c^{3} \sqrt {- \frac {a^{3}}{c^{7}}}}{a} \right )}}{8} + \frac {x^{6}}{6 c^{2}} \] Input:
integrate(x**13/(c*x**4+a)**2,x)
Output:
-a**2*x**2/(4*a*c**3 + 4*c**4*x**4) - a*x**2/c**3 - 5*sqrt(-a**3/c**7)*log (x**2 - c**3*sqrt(-a**3/c**7)/a)/8 + 5*sqrt(-a**3/c**7)*log(x**2 + c**3*sq rt(-a**3/c**7)/a)/8 + x**6/(6*c**2)
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=-\frac {a^{2} x^{2}}{4 \, {\left (c^{4} x^{4} + a c^{3}\right )}} + \frac {5 \, a^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c^{3}} + \frac {c x^{6} - 6 \, a x^{2}}{6 \, c^{3}} \] Input:
integrate(x^13/(c*x^4+a)^2,x, algorithm="maxima")
Output:
-1/4*a^2*x^2/(c^4*x^4 + a*c^3) + 5/4*a^2*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c )*c^3) + 1/6*(c*x^6 - 6*a*x^2)/c^3
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=-\frac {a^{2} x^{2}}{4 \, {\left (c x^{4} + a\right )} c^{3}} + \frac {5 \, a^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c^{3}} + \frac {c^{4} x^{6} - 6 \, a c^{3} x^{2}}{6 \, c^{6}} \] Input:
integrate(x^13/(c*x^4+a)^2,x, algorithm="giac")
Output:
-1/4*a^2*x^2/((c*x^4 + a)*c^3) + 5/4*a^2*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c )*c^3) + 1/6*(c^4*x^6 - 6*a*c^3*x^2)/c^6
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=\frac {x^6}{6\,c^2}-\frac {a^2\,x^2}{4\,\left (c^4\,x^4+a\,c^3\right )}-\frac {a\,x^2}{c^3}+\frac {5\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,c^{7/2}} \] Input:
int(x^13/(a + c*x^4)^2,x)
Output:
x^6/(6*c^2) - (a^2*x^2)/(4*(a*c^3 + c^4*x^4)) - (a*x^2)/c^3 + (5*a^(3/2)*a tan((c^(1/2)*x^2)/a^(1/2)))/(4*c^(7/2))
Time = 0.18 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.66 \[ \int \frac {x^{13}}{\left (a+c x^4\right )^2} \, dx=\frac {-15 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-15 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-15 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-15 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-15 a^{2} c \,x^{2}-10 a \,c^{2} x^{6}+2 c^{3} x^{10}}{12 c^{4} \left (c \,x^{4}+a \right )} \] Input:
int(x^13/(c*x^4+a)^2,x)
Output:
( - 15*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 - 15*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 - 15*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 - 15*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*s qrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**4 - 15*a**2*c*x**2 - 10*a*c* *2*x**6 + 2*c**3*x**10)/(12*c**4*(a + c*x**4))