Integrand size = 22, antiderivative size = 261 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\frac {C x}{c^2}-\frac {\left (a B c-A c^2-a^2 C\right ) x}{6 a c^2 \left (a+c x^6\right )}+\frac {\left (a B c+5 A c^2-7 a^2 C\right ) \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} c^{13/6}}-\frac {\left (a B c+5 A c^2-7 a^2 C\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} c^{13/6}}+\frac {\left (a B c+5 A c^2-7 a^2 C\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} c^{13/6}}+\frac {\left (a B c+5 A c^2-7 a^2 C\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{12 \sqrt {3} a^{11/6} c^{13/6}} \] Output:
C*x/c^2-1/6*(-A*c^2+B*a*c-C*a^2)*x/a/c^2/(c*x^6+a)+1/18*(5*A*c^2+B*a*c-7*C *a^2)*arctan(c^(1/6)*x/a^(1/6))/a^(11/6)/c^(13/6)+1/36*(5*A*c^2+B*a*c-7*C* a^2)*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(11/6)/c^(13/6)+1/36*(5*A*c^2+ B*a*c-7*C*a^2)*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(11/6)/c^(13/6)+1/36* (5*A*c^2+B*a*c-7*C*a^2)*arctanh(3^(1/2)*a^(1/6)*c^(1/6)*x/(a^(1/3)+c^(1/3) *x^2))*3^(1/2)/a^(11/6)/c^(13/6)
Time = 0.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\frac {72 \sqrt [6]{c} C x+\frac {12 \sqrt [6]{c} \left (-a B c+A c^2+a^2 C\right ) x}{a \left (a+c x^6\right )}-\frac {4 \left (-a B c-5 A c^2+7 a^2 C\right ) \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{11/6}}+\frac {2 \left (-a B c-5 A c^2+7 a^2 C\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{11/6}}-\frac {2 \left (-a B c-5 A c^2+7 a^2 C\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{11/6}}+\frac {\sqrt {3} \left (-a B c-5 A c^2+7 a^2 C\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{11/6}}-\frac {\sqrt {3} \left (-a B c-5 A c^2+7 a^2 C\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{11/6}}}{72 c^{13/6}} \] Input:
Integrate[(A + B*x^6 + C*x^12)/(a + c*x^6)^2,x]
Output:
(72*c^(1/6)*C*x + (12*c^(1/6)*(-(a*B*c) + A*c^2 + a^2*C)*x)/(a*(a + c*x^6) ) - (4*(-(a*B*c) - 5*A*c^2 + 7*a^2*C)*ArcTan[(c^(1/6)*x)/a^(1/6)])/a^(11/6 ) + (2*(-(a*B*c) - 5*A*c^2 + 7*a^2*C)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/ 6)])/a^(11/6) - (2*(-(a*B*c) - 5*A*c^2 + 7*a^2*C)*ArcTan[Sqrt[3] + (2*c^(1 /6)*x)/a^(1/6)])/a^(11/6) + (Sqrt[3]*(-(a*B*c) - 5*A*c^2 + 7*a^2*C)*Log[a^ (1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/a^(11/6) - (Sqrt[3]*(-(a *B*c) - 5*A*c^2 + 7*a^2*C)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/ 3)*x^2])/a^(11/6))/(72*c^(13/6))
Time = 0.88 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1739, 25, 913, 753, 27, 218, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 1739 |
| \(\displaystyle -\frac {\int -\frac {6 a c C x^6+5 A c^2+a B c-a^2 C}{c x^6+a}dx}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 a c C x^6+5 A c^2+a B c-a^2 C}{c x^6+a}dx}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \int \frac {1}{c x^6+a}dx+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{c} x}{2 \left (\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{c} x+2 \sqrt [6]{a}}{2 \left (\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{c} x+2 \sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{c} x+2 \sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{c}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{c}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (-7 a^2 C+a B c+5 A c^2\right ) \left (\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{c}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{c}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+6 a C x}{6 a c^2}-\frac {x \left (a^2 (-C)+a B c-A c^2\right )}{6 a c^2 \left (a+c x^6\right )}\) |
Input:
Int[(A + B*x^6 + C*x^12)/(a + c*x^6)^2,x]
Output:
-1/6*((a*B*c - A*c^2 - a^2*C)*x)/(a*c^2*(a + c*x^6)) + (6*a*C*x + (a*B*c + 5*A*c^2 - 7*a^2*C)*(ArcTan[(c^(1/6)*x)/a^(1/6)]/(3*a^(5/6)*c^(1/6)) + (-( ArcTan[Sqrt[3]*(1 - (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/c^(1/6)) - (Sqrt[3]* Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(2*c^(1/6)))/(6*a^ (5/6)) + (ArcTan[Sqrt[3]*(1 + (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/c^(1/6) + (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(2*c^(1/6 )))/(6*a^(5/6))))/(6*a*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[(-(c*d^2 - b*d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d* e^2*n*(q + 1))), x] + Simp[1/(n*(q + 1)*d*e^2) Int[(d + e*x^n)^(q + 1)*Si mp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && N eQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.34
| method | result | size |
| risch | \(\frac {C x}{c^{2}}+\frac {\left (A \,c^{2}-a B c +a^{2} C \right ) x}{6 a \,c^{2} \left (c \,x^{6}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+a \right )}{\sum }\frac {\left (5 A \,c^{2}+a B c -7 a^{2} C \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{36 c^{3} a}\) | \(88\) |
| default | \(\frac {C x}{c^{2}}+\frac {\frac {\left (A \,c^{2}-a B c +a^{2} C \right ) x}{6 a \left (c \,x^{6}+a \right )}+\frac {\left (5 A \,c^{2}+a B c -7 a^{2} C \right ) \left (-\frac {\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{6 a}}{c^{2}}\) | \(224\) |
Input:
int((C*x^12+B*x^6+A)/(c*x^6+a)^2,x,method=_RETURNVERBOSE)
Output:
C*x/c^2+1/6*(A*c^2-B*a*c+C*a^2)/a*x/c^2/(c*x^6+a)+1/36/c^3/a*sum((5*A*c^2+ B*a*c-7*C*a^2)/_R^5*ln(x-_R),_R=RootOf(_Z^6*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 4144 vs. \(2 (212) = 424\).
Time = 0.12 (sec) , antiderivative size = 4144, normalized size of antiderivative = 15.88 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((C*x^12+B*x^6+A)/(c*x^6+a)^2,x, algorithm="fricas")
Output:
Too large to include
Time = 78.07 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.90 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\frac {C x}{c^{2}} + \frac {x \left (A c^{2} - B a c + C a^{2}\right )}{6 a^{2} c^{2} + 6 a c^{3} x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{11} c^{13} + 15625 A^{6} c^{12} + 18750 A^{5} B a c^{11} - 131250 A^{5} C a^{2} c^{10} + 9375 A^{4} B^{2} a^{2} c^{10} - 131250 A^{4} B C a^{3} c^{9} + 459375 A^{4} C^{2} a^{4} c^{8} + 2500 A^{3} B^{3} a^{3} c^{9} - 52500 A^{3} B^{2} C a^{4} c^{8} + 367500 A^{3} B C^{2} a^{5} c^{7} - 857500 A^{3} C^{3} a^{6} c^{6} + 375 A^{2} B^{4} a^{4} c^{8} - 10500 A^{2} B^{3} C a^{5} c^{7} + 110250 A^{2} B^{2} C^{2} a^{6} c^{6} - 514500 A^{2} B C^{3} a^{7} c^{5} + 900375 A^{2} C^{4} a^{8} c^{4} + 30 A B^{5} a^{5} c^{7} - 1050 A B^{4} C a^{6} c^{6} + 14700 A B^{3} C^{2} a^{7} c^{5} - 102900 A B^{2} C^{3} a^{8} c^{4} + 360150 A B C^{4} a^{9} c^{3} - 504210 A C^{5} a^{10} c^{2} + B^{6} a^{6} c^{6} - 42 B^{5} C a^{7} c^{5} + 735 B^{4} C^{2} a^{8} c^{4} - 6860 B^{3} C^{3} a^{9} c^{3} + 36015 B^{2} C^{4} a^{10} c^{2} - 100842 B C^{5} a^{11} c + 117649 C^{6} a^{12}, \left ( t \mapsto t \log {\left (- \frac {36 t a^{2} c^{2}}{- 5 A c^{2} - B a c + 7 C a^{2}} + x \right )} \right )\right )} \] Input:
integrate((C*x**12+B*x**6+A)/(c*x**6+a)**2,x)
Output:
C*x/c**2 + x*(A*c**2 - B*a*c + C*a**2)/(6*a**2*c**2 + 6*a*c**3*x**6) + Roo tSum(2176782336*_t**6*a**11*c**13 + 15625*A**6*c**12 + 18750*A**5*B*a*c**1 1 - 131250*A**5*C*a**2*c**10 + 9375*A**4*B**2*a**2*c**10 - 131250*A**4*B*C *a**3*c**9 + 459375*A**4*C**2*a**4*c**8 + 2500*A**3*B**3*a**3*c**9 - 52500 *A**3*B**2*C*a**4*c**8 + 367500*A**3*B*C**2*a**5*c**7 - 857500*A**3*C**3*a **6*c**6 + 375*A**2*B**4*a**4*c**8 - 10500*A**2*B**3*C*a**5*c**7 + 110250* A**2*B**2*C**2*a**6*c**6 - 514500*A**2*B*C**3*a**7*c**5 + 900375*A**2*C**4 *a**8*c**4 + 30*A*B**5*a**5*c**7 - 1050*A*B**4*C*a**6*c**6 + 14700*A*B**3* C**2*a**7*c**5 - 102900*A*B**2*C**3*a**8*c**4 + 360150*A*B*C**4*a**9*c**3 - 504210*A*C**5*a**10*c**2 + B**6*a**6*c**6 - 42*B**5*C*a**7*c**5 + 735*B* *4*C**2*a**8*c**4 - 6860*B**3*C**3*a**9*c**3 + 36015*B**2*C**4*a**10*c**2 - 100842*B*C**5*a**11*c + 117649*C**6*a**12, Lambda(_t, _t*log(-36*_t*a**2 *c**2/(-5*A*c**2 - B*a*c + 7*C*a**2) + x)))
Time = 0.12 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\frac {{\left (C a^{2} - B a c + A c^{2}\right )} x}{6 \, {\left (a c^{3} x^{6} + a^{2} c^{2}\right )}} + \frac {C x}{c^{2}} - \frac {\frac {\sqrt {3} {\left (7 \, C a^{2} - B a c - 5 \, A c^{2}\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} c^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (7 \, C a^{2} - B a c - 5 \, A c^{2}\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} c^{\frac {1}{6}}} + \frac {4 \, {\left (7 \, C a^{2} c^{\frac {1}{3}} - B a c^{\frac {4}{3}} - 5 \, A c^{\frac {7}{3}}\right )} \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, C a^{\frac {7}{3}} c^{\frac {1}{3}} - B a^{\frac {4}{3}} c^{\frac {4}{3}} - 5 \, A a^{\frac {1}{3}} c^{\frac {7}{3}}\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, C a^{\frac {7}{3}} c^{\frac {1}{3}} - B a^{\frac {4}{3}} c^{\frac {4}{3}} - 5 \, A a^{\frac {1}{3}} c^{\frac {7}{3}}\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{72 \, a c^{2}} \] Input:
integrate((C*x^12+B*x^6+A)/(c*x^6+a)^2,x, algorithm="maxima")
Output:
1/6*(C*a^2 - B*a*c + A*c^2)*x/(a*c^3*x^6 + a^2*c^2) + C*x/c^2 - 1/72*(sqrt (3)*(7*C*a^2 - B*a*c - 5*A*c^2)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c^(1/6)* x + a^(1/3))/(a^(5/6)*c^(1/6)) - sqrt(3)*(7*C*a^2 - B*a*c - 5*A*c^2)*log(c ^(1/3)*x^2 - sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a^(5/6)*c^(1/6)) + 4*(7 *C*a^2*c^(1/3) - B*a*c^(4/3) - 5*A*c^(7/3))*arctan(c^(1/3)*x/sqrt(a^(1/3)* c^(1/3)))/(a^(2/3)*c^(1/3)*sqrt(a^(1/3)*c^(1/3))) + 2*(7*C*a^(7/3)*c^(1/3) - B*a^(4/3)*c^(4/3) - 5*A*a^(1/3)*c^(7/3))*arctan((2*c^(1/3)*x + sqrt(3)* a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(1/3)*sqrt(a^(1/3)*c^(1/3))) + 2*(7*C*a^(7/3)*c^(1/3) - B*a^(4/3)*c^(4/3) - 5*A*a^(1/3)*c^(7/3))*arctan ((2*c^(1/3)*x - sqrt(3)*a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(1/3) *sqrt(a^(1/3)*c^(1/3))))/(a*c^2)
Time = 0.12 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\frac {C x}{c^{2}} - \frac {\sqrt {3} {\left (7 \, \left (a c^{5}\right )^{\frac {1}{6}} C a^{2} - \left (a c^{5}\right )^{\frac {1}{6}} B a c - 5 \, \left (a c^{5}\right )^{\frac {1}{6}} A c^{2}\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} c^{3}} + \frac {\sqrt {3} {\left (7 \, \left (a c^{5}\right )^{\frac {1}{6}} C a^{2} - \left (a c^{5}\right )^{\frac {1}{6}} B a c - 5 \, \left (a c^{5}\right )^{\frac {1}{6}} A c^{2}\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} c^{3}} + \frac {C a^{2} x - B a c x + A c^{2} x}{6 \, {\left (c x^{6} + a\right )} a c^{2}} - \frac {{\left (7 \, \left (a c^{5}\right )^{\frac {1}{6}} C a^{2} - \left (a c^{5}\right )^{\frac {1}{6}} B a c - 5 \, \left (a c^{5}\right )^{\frac {1}{6}} A c^{2}\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} c^{3}} - \frac {{\left (7 \, \left (a c^{5}\right )^{\frac {1}{6}} C a^{2} - \left (a c^{5}\right )^{\frac {1}{6}} B a c - 5 \, \left (a c^{5}\right )^{\frac {1}{6}} A c^{2}\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} c^{3}} - \frac {{\left (7 \, \left (a c^{5}\right )^{\frac {1}{6}} C a^{2} - \left (a c^{5}\right )^{\frac {1}{6}} B a c - 5 \, \left (a c^{5}\right )^{\frac {1}{6}} A c^{2}\right )} \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} c^{3}} \] Input:
integrate((C*x^12+B*x^6+A)/(c*x^6+a)^2,x, algorithm="giac")
Output:
C*x/c^2 - 1/72*sqrt(3)*(7*(a*c^5)^(1/6)*C*a^2 - (a*c^5)^(1/6)*B*a*c - 5*(a *c^5)^(1/6)*A*c^2)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^3 ) + 1/72*sqrt(3)*(7*(a*c^5)^(1/6)*C*a^2 - (a*c^5)^(1/6)*B*a*c - 5*(a*c^5)^ (1/6)*A*c^2)*log(x^2 - sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^3) + 1/ 6*(C*a^2*x - B*a*c*x + A*c^2*x)/((c*x^6 + a)*a*c^2) - 1/36*(7*(a*c^5)^(1/6 )*C*a^2 - (a*c^5)^(1/6)*B*a*c - 5*(a*c^5)^(1/6)*A*c^2)*arctan((2*x + sqrt( 3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^3) - 1/36*(7*(a*c^5)^(1/6)*C*a^2 - (a* c^5)^(1/6)*B*a*c - 5*(a*c^5)^(1/6)*A*c^2)*arctan((2*x - sqrt(3)*(a/c)^(1/6 ))/(a/c)^(1/6))/(a^2*c^3) - 1/18*(7*(a*c^5)^(1/6)*C*a^2 - (a*c^5)^(1/6)*B* a*c - 5*(a*c^5)^(1/6)*A*c^2)*arctan(x/(a/c)^(1/6))/(a^2*c^3)
Time = 6.43 (sec) , antiderivative size = 4448, normalized size of antiderivative = 17.04 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx=\text {Too large to display} \] Input:
int((A + B*x^6 + C*x^12)/(a + c*x^6)^2,x)
Output:
(C*x)/c^2 + (atan(((((x*(625*A^4*c^8 + 2401*C^4*a^8 + B^4*a^4*c^4 + 150*A^ 2*B^2*a^2*c^6 + 7350*A^2*C^2*a^4*c^4 + 294*B^2*C^2*a^6*c^2 + 500*A^3*B*a*c ^7 - 1372*B*C^3*a^7*c + 20*A*B^3*a^3*c^5 - 6860*A*C^3*a^6*c^2 - 3500*A^3*C *a^2*c^6 - 28*B^3*C*a^5*c^3 + 2940*A*B*C^2*a^5*c^3 - 420*A*B^2*C*a^4*c^4 - 2100*A^2*B*C*a^3*c^5))/(216*a^4*c^3) - ((5*A*c^2 - 7*C*a^2 + B*a*c)*(125* A^3*c^6 - 343*C^3*a^6 + B^3*a^3*c^3 + 75*A^2*B*a*c^5 + 147*B*C^2*a^5*c + 1 5*A*B^2*a^2*c^4 + 735*A*C^2*a^4*c^2 - 525*A^2*C*a^2*c^4 - 21*B^2*C*a^4*c^2 - 210*A*B*C*a^3*c^3))/(216*(-a)^(23/6)*c^(19/6)))*(5*A*c^2 - 7*C*a^2 + B* a*c)*1i)/(36*(-a)^(11/6)*c^(13/6)) + (((x*(625*A^4*c^8 + 2401*C^4*a^8 + B^ 4*a^4*c^4 + 150*A^2*B^2*a^2*c^6 + 7350*A^2*C^2*a^4*c^4 + 294*B^2*C^2*a^6*c ^2 + 500*A^3*B*a*c^7 - 1372*B*C^3*a^7*c + 20*A*B^3*a^3*c^5 - 6860*A*C^3*a^ 6*c^2 - 3500*A^3*C*a^2*c^6 - 28*B^3*C*a^5*c^3 + 2940*A*B*C^2*a^5*c^3 - 420 *A*B^2*C*a^4*c^4 - 2100*A^2*B*C*a^3*c^5))/(216*a^4*c^3) + ((5*A*c^2 - 7*C* a^2 + B*a*c)*(125*A^3*c^6 - 343*C^3*a^6 + B^3*a^3*c^3 + 75*A^2*B*a*c^5 + 1 47*B*C^2*a^5*c + 15*A*B^2*a^2*c^4 + 735*A*C^2*a^4*c^2 - 525*A^2*C*a^2*c^4 - 21*B^2*C*a^4*c^2 - 210*A*B*C*a^3*c^3))/(216*(-a)^(23/6)*c^(19/6)))*(5*A* c^2 - 7*C*a^2 + B*a*c)*1i)/(36*(-a)^(11/6)*c^(13/6)))/((((x*(625*A^4*c^8 + 2401*C^4*a^8 + B^4*a^4*c^4 + 150*A^2*B^2*a^2*c^6 + 7350*A^2*C^2*a^4*c^4 + 294*B^2*C^2*a^6*c^2 + 500*A^3*B*a*c^7 - 1372*B*C^3*a^7*c + 20*A*B^3*a^3*c ^5 - 6860*A*C^3*a^6*c^2 - 3500*A^3*C*a^2*c^6 - 28*B^3*C*a^5*c^3 + 2940*...
Time = 0.18 (sec) , antiderivative size = 980, normalized size of antiderivative = 3.75 \[ \int \frac {A+B x^6+C x^{12}}{\left (a+c x^6\right )^2} \, dx =\text {Too large to display} \] Input:
int((C*x^12+B*x^6+A)/(c*x^6+a)^2,x)
Output:
(14*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c** (1/6)*a**(1/6)))*a**2 - 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3 ) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*b + 14*c**(1/6)*a**(1/6)*atan((c* *(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c*x**6 - 10 *c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/ 6)*a**(1/6)))*a*c - 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*b*c*x**6 - 10*c**(1/6)*a**(1/6)*atan((c **(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c**2*x**6 - 14*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**( 1/6)*a**(1/6)))*a**2 + 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*b - 14*c**(1/6)*a**(1/6)*atan((c** (1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c*x**6 + 10* c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6 )*a**(1/6)))*a*c + 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2 *c**(1/3)*x)/(c**(1/6)*a**(1/6)))*b*c*x**6 + 10*c**(1/6)*a**(1/6)*atan((c* *(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c**2*x**6 - 2 8*c**(1/6)*a**(1/6)*atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a**2 + 4*c**(1/ 6)*a**(1/6)*atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*b - 28*c**(1/6)*a**(1 /6)*atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c*x**6 + 20*c**(1/6)*a**(1/6) *atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c + 4*c**(1/6)*a**(1/6)*atan(...