Integrand size = 14, antiderivative size = 174 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=-\frac {3 b c}{4 x}-\frac {1}{4} b c^{4/3} \arctan \left (\sqrt [3]{c} x\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}+\frac {1}{8} b c^{4/3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right ) \]
-3/4*b*c/x-1/4*b*c^(4/3)*arctan(c^(1/3)*x)+1/4*(-a-b*arctan(c*x^3))/x^4-1/ 8*b*c^(4/3)*arctan(2*c^(1/3)*x-3^(1/2))-1/8*b*c^(4/3)*arctan(2*c^(1/3)*x+3 ^(1/2))-1/16*b*c^(4/3)*ln(1+c^(2/3)*x^2-c^(1/3)*x*3^(1/2))*3^(1/2)+1/16*b* c^(4/3)*ln(1+c^(2/3)*x^2+c^(1/3)*x*3^(1/2))*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {3 b c}{4 x}-\frac {1}{4} b c^{4/3} \arctan \left (\sqrt [3]{c} x\right )-\frac {b \arctan \left (c x^3\right )}{4 x^4}+\frac {1}{8} b c^{4/3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right ) \]
-1/4*a/x^4 - (3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (b*ArcTan[c *x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/8 - (b*c^(4/3)* ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1 /3)*x + c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16
Time = 0.40 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5361, 847, 824, 27, 216, 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.
| \(\displaystyle \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3}{4} b c \int \frac {1}{x^2 \left (c^2 x^6+1\right )}dx-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\int \frac {x^4}{c^2 x^6+1}dx\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (\frac {\int \frac {1}{c^{2/3} x^2+1}dx}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt {3} \sqrt [3]{c} x}{2 \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [3]{c} x+1}{2 \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (\frac {\int \frac {1}{c^{2/3} x^2+1}dx}{3 c^{4/3}}-\frac {\int \frac {1-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {\sqrt {3} \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\int \frac {1-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {\sqrt {3} \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \int -\frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\int \frac {1}{-\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )}{\sqrt {3} \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}-\frac {\sqrt {3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\right )-\frac {1}{x}\right )-\frac {a+b \arctan \left (c x^3\right )}{4 x^4}\) |
-1/4*(a + b*ArcTan[c*x^3])/x^4 + (3*b*c*(-x^(-1) - c^2*(ArcTan[c^(1/3)*x]/ (3*c^(5/3)) - (ArcTan[Sqrt[3]*(1 - (2*c^(1/3)*x)/Sqrt[3])]/c^(1/3) - (Sqrt [3]*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(2*c^(1/3)))/(6*c^(4/3)) - ( -(ArcTan[Sqrt[3]*(1 + (2*c^(1/3)*x)/Sqrt[3])]/c^(1/3)) + (Sqrt[3]*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(2*c^(1/3)))/(6*c^(4/3)))))/4
3.2.12.3.1 Defintions of rubi rules used
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Time = 0.64 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {a}{4 x^{4}}+b \left (-\frac {\arctan \left (c \,x^{3}\right )}{4 x^{4}}+\frac {3 c \left (-\frac {1}{x}-\left (-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) c^{2}\right )}{4}\right )\) | \(164\) |
| parts | \(-\frac {a}{4 x^{4}}+b \left (-\frac {\arctan \left (c \,x^{3}\right )}{4 x^{4}}+\frac {3 c \left (-\frac {1}{x}-\left (-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) c^{2}\right )}{4}\right )\) | \(164\) |
-1/4*a/x^4+b*(-1/4/x^4*arctan(c*x^3)+3/4*c*(-1/x-(-1/12*3^(1/2)*(1/c^2)^(5 /6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))+1/6/c^2/(1/c^2)^(1/6)*ar ctan(2*x/(1/c^2)^(1/6)+3^(1/2))+1/12*3^(1/2)*(1/c^2)^(5/6)*ln(x^2-3^(1/2)* (1/c^2)^(1/6)*x+(1/c^2)^(1/3))+1/6/c^2/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1 /6)-3^(1/2))+1/3/c^2/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6)))*c^2))
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (126) = 252\).
Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=-\frac {2 \, \left (-b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (b^{5} c^{7} x + \left (-b^{6} c^{8}\right )^{\frac {5}{6}}\right ) - 2 \, \left (-b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (b^{5} c^{7} x - \left (-b^{6} c^{8}\right )^{\frac {5}{6}}\right ) + 12 \, b c x^{3} - \left (-b^{6} c^{8}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} \log \left (2 \, b^{5} c^{7} x + \left (-b^{6} c^{8}\right )^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )}\right ) + \left (-b^{6} c^{8}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} \log \left (2 \, b^{5} c^{7} x - \left (-b^{6} c^{8}\right )^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )}\right ) - \left (-b^{6} c^{8}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} \log \left (2 \, b^{5} c^{7} x + \left (-b^{6} c^{8}\right )^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )}\right ) + \left (-b^{6} c^{8}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} \log \left (2 \, b^{5} c^{7} x - \left (-b^{6} c^{8}\right )^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )}\right ) + 4 \, b \arctan \left (c x^{3}\right ) + 4 \, a}{16 \, x^{4}} \]
-1/16*(2*(-b^6*c^8)^(1/6)*x^4*log(b^5*c^7*x + (-b^6*c^8)^(5/6)) - 2*(-b^6* c^8)^(1/6)*x^4*log(b^5*c^7*x - (-b^6*c^8)^(5/6)) + 12*b*c*x^3 - (-b^6*c^8) ^(1/6)*(sqrt(-3)*x^4 - x^4)*log(2*b^5*c^7*x + (-b^6*c^8)^(5/6)*(sqrt(-3) + 1)) + (-b^6*c^8)^(1/6)*(sqrt(-3)*x^4 - x^4)*log(2*b^5*c^7*x - (-b^6*c^8)^ (5/6)*(sqrt(-3) + 1)) - (-b^6*c^8)^(1/6)*(sqrt(-3)*x^4 + x^4)*log(2*b^5*c^ 7*x + (-b^6*c^8)^(5/6)*(sqrt(-3) - 1)) + (-b^6*c^8)^(1/6)*(sqrt(-3)*x^4 + x^4)*log(2*b^5*c^7*x - (-b^6*c^8)^(5/6)*(sqrt(-3) - 1)) + 4*b*arctan(c*x^3 ) + 4*a)/x^4
Time = 49.05 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=\begin {cases} - \frac {a}{4 x^{4}} + \frac {3 b c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}} \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{16} - \frac {3 b c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{16} + \frac {\sqrt {3} b c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{8} + \frac {\sqrt {3} b c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{8} - \frac {b c^{2} \sqrt [3]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{4} - \frac {3 b c}{4 x} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a}{4 x^{4}} & \text {otherwise} \end {cases} \]
Piecewise((-a/(4*x**4) + 3*b*c**3*(-1/c**2)**(5/6)*log(4*x**2 - 4*x*(-1/c* *2)**(1/6) + 4*(-1/c**2)**(1/3))/16 - 3*b*c**3*(-1/c**2)**(5/6)*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/16 + sqrt(3)*b*c**3*(-1/c**2 )**(5/6)*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) - sqrt(3)/3)/8 + sqrt(3)*b* c**3*(-1/c**2)**(5/6)*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) + sqrt(3)/3)/8 - b*c**2*(-1/c**2)**(1/3)*atan(c*x**3)/4 - 3*b*c/(4*x) - b*atan(c*x**3)/( 4*x**4), Ne(c, 0)), (-a/(4*x**4), True))
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=\frac {1}{16} \, {\left ({\left (c^{2} {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )} - \frac {12}{x}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
1/16*((c^2*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - sqr t(3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - 4*arctan(c^(1/3)*x )/c^(5/3) - 2*arctan((2*c^(2/3)*x + sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3) - 2* arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3)) - 12/x)*c - 4*arc tan(c*x^3)/x^4)*b - 1/4*a/x^4
Time = 0.41 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=\frac {1}{16} \, b c^{3} {\left (\frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} - \frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} - \frac {4 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}}\right )} - \frac {3 \, b c x^{3} + b \arctan \left (c x^{3}\right ) + a}{4 \, x^{4}} \]
1/16*b*c^3*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs( c)^(2/3))/c^2 - sqrt(3)*abs(c)^(1/3)*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/ abs(c)^(2/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*abs (c)^(1/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c) ^(1/3))/c^2 - 4*abs(c)^(1/3)*arctan(x*abs(c)^(1/3))/c^2) - 1/4*(3*b*c*x^3 + b*arctan(c*x^3) + a)/x^4
Time = 0.79 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^5} \, dx=-\frac {a}{4\,x^4}+\frac {b\,c^{4/3}\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{8}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{4\,x^4}-\frac {3\,b\,c}{4\,x}-\frac {\sqrt {3}\,b\,c^{4/3}\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )\,1{}\mathrm {i}}{8} \]
(b*c^(4/3)*(atan((-1)^(2/3)*c^(1/3)*x) + atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/ 2)*1i - 1))/2) + 2*atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i + 1))/2)))/8 - a /(4*x^4) - (b*atan(c*x^3))/(4*x^4) - (3*b*c)/(4*x) - (3^(1/2)*b*c^(4/3)*(a tan((-1)^(2/3)*c^(1/3)*x) - atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i - 1))/2 ))*1i)/8