Integrand size = 16, antiderivative size = 87 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1+c^2 x^6\right ) \]
-1/3*b*c*(a+b*arctan(c*x^3))/x^3-1/6*c^2*(a+b*arctan(c*x^3))^2-1/6*(a+b*ar ctan(c*x^3))^2/x^6+b^2*c^2*ln(x)-1/6*b^2*c^2*ln(c^2*x^6+1)
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {a^2+2 a b c x^3+2 b \left (a+b c x^3+a c^2 x^6\right ) \arctan \left (c x^3\right )+b^2 \left (1+c^2 x^6\right ) \arctan \left (c x^3\right )^2-6 b^2 c^2 x^6 \log (x)+b^2 c^2 x^6 \log \left (1+c^2 x^6\right )}{6 x^6} \]
-1/6*(a^2 + 2*a*b*c*x^3 + 2*b*(a + b*c*x^3 + a*c^2*x^6)*ArcTan[c*x^3] + b^ 2*(1 + c^2*x^6)*ArcTan[c*x^3]^2 - 6*b^2*c^2*x^6*Log[x] + b^2*c^2*x^6*Log[1 + c^2*x^6])/x^6
Time = 0.50 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5363, 5361, 5453, 5361, 243, 47, 14, 16, 5419}
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.
| \(\displaystyle \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^9}dx^3\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} \left (b c \int \frac {a+b \arctan \left (c x^3\right )}{x^6 \left (c^2 x^6+1\right )}dx^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{3} \left (b c \left (\int \frac {a+b \arctan \left (c x^3\right )}{x^6}dx^3-c^2 \int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} \left (b c \left (c^2 \left (-\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )+b c \int \frac {1}{x^3 \left (c^2 x^6+1\right )}dx^3-\frac {a+b \arctan \left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{3} \left (b c \left (c^2 \left (-\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )+\frac {1}{2} b c \int \frac {1}{x^3 \left (c^2 x^6+1\right )}dx^6-\frac {a+b \arctan \left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {1}{3} \left (b c \left (c^2 \left (-\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )+\frac {1}{2} b c \left (\int \frac {1}{x^3}dx^6-c^2 \int \frac {1}{c^2 x^6+1}dx^6\right )-\frac {a+b \arctan \left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {1}{3} \left (b c \left (c^2 \left (-\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )+\frac {1}{2} b c \left (\log \left (x^6\right )-c^2 \int \frac {1}{c^2 x^6+1}dx^6\right )-\frac {a+b \arctan \left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (b c \left (c^2 \left (-\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3\right )-\frac {a+b \arctan \left (c x^3\right )}{x^3}+\frac {1}{2} b c \left (\log \left (x^6\right )-\log \left (c^2 x^6+1\right )\right )\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{3} \left (b c \left (-\frac {c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 b}-\frac {a+b \arctan \left (c x^3\right )}{x^3}+\frac {1}{2} b c \left (\log \left (x^6\right )-\log \left (c^2 x^6+1\right )\right )\right )-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^6}\right )\) |
(-1/2*(a + b*ArcTan[c*x^3])^2/x^6 + b*c*(-((a + b*ArcTan[c*x^3])/x^3) - (c *(a + b*ArcTan[c*x^3])^2)/(2*b) + (b*c*(Log[x^6] - Log[1 + c^2*x^6]))/2))/ 3
3.2.19.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplif y[(m + 1)/n]]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Time = 0.78 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(-\frac {a^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2} c^{2}}{6}-\frac {b^{2} c \arctan \left (c \,x^{3}\right )}{3 x^{3}}+b^{2} c^{2} \ln \left (x \right )-\frac {b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right )}{6}-\frac {a b \arctan \left (c \,x^{3}\right )}{3 x^{6}}-\frac {a b \arctan \left (c \,x^{3}\right ) c^{2}}{3}-\frac {a b c}{3 x^{3}}\) | \(118\) |
| parts | \(-\frac {a^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2} c^{2}}{6}-\frac {b^{2} c \arctan \left (c \,x^{3}\right )}{3 x^{3}}+b^{2} c^{2} \ln \left (x \right )-\frac {b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right )}{6}-\frac {a b \arctan \left (c \,x^{3}\right )}{3 x^{6}}-\frac {a b \arctan \left (c \,x^{3}\right ) c^{2}}{3}-\frac {a b c}{3 x^{3}}\) | \(118\) |
| parallelrisch | \(\frac {-b^{2} \arctan \left (c \,x^{3}\right )^{2} x^{6} c^{2}+6 b^{2} c^{2} \ln \left (x \right ) x^{6}-b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right ) x^{6}-2 a b \arctan \left (c \,x^{3}\right ) x^{6} c^{2}+a^{2} c^{2} x^{6}-2 b^{2} \arctan \left (c \,x^{3}\right ) x^{3} c -2 a b c \,x^{3}-b^{2} \arctan \left (c \,x^{3}\right )^{2}-2 a b \arctan \left (c \,x^{3}\right )-a^{2}}{6 x^{6}}\) | \(137\) |
| risch | \(\frac {b^{2} \left (c^{2} x^{6}+1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{24 x^{6}}+\frac {i b \left (i b \,c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )+2 b c \,x^{3}+2 a +i b \ln \left (-i c \,x^{3}+1\right )\right ) \ln \left (i c \,x^{3}+1\right )}{12 x^{6}}-\frac {4 i \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) a b \,c^{2} x^{6}-4 i \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) a b \,c^{2} x^{6}-b^{2} c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )^{2}-24 b^{2} c^{2} \ln \left (x \right ) x^{6}+4 \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) b^{2} c^{2} x^{6}+4 \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) b^{2} c^{2} x^{6}+4 i b^{2} c \,x^{3} \ln \left (-i c \,x^{3}+1\right )+8 a b c \,x^{3}+4 i b \ln \left (-i c \,x^{3}+1\right ) a -b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}+4 a^{2}}{24 x^{6}}\) | \(332\) |
-1/6*a^2/x^6-1/6*b^2/x^6*arctan(c*x^3)^2-1/6*b^2*arctan(c*x^3)^2*c^2-1/3*b ^2*c*arctan(c*x^3)/x^3+b^2*c^2*ln(x)-1/6*b^2*c^2*ln(c^2*x^6+1)-1/3*a*b/x^6 *arctan(c*x^3)-1/3*a*b*arctan(c*x^3)*c^2-1/3*a*b*c/x^3
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {b^{2} c^{2} x^{6} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 2 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} + b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + a^{2} + 2 \, {\left (a b c^{2} x^{6} + b^{2} c x^{3} + a b\right )} \arctan \left (c x^{3}\right )}{6 \, x^{6}} \]
-1/6*(b^2*c^2*x^6*log(c^2*x^6 + 1) - 6*b^2*c^2*x^6*log(x) + 2*a*b*c*x^3 + (b^2*c^2*x^6 + b^2)*arctan(c*x^3)^2 + a^2 + 2*(a*b*c^2*x^6 + b^2*c*x^3 + a *b)*arctan(c*x^3))/x^6
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (80) = 160\).
Time = 71.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=\begin {cases} - \frac {a^{2}}{6 x^{6}} - \frac {a b c^{2} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {a b c}{3 x^{3}} - \frac {a b \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{6}} + \frac {b^{2} c^{3} \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{3} + b^{2} c^{2} \log {\left (x \right )} - \frac {b^{2} c^{2} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6} - \frac {b^{2} c \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{3}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{6 x^{6}} & \text {otherwise} \end {cases} \]
Piecewise((-a**2/(6*x**6) - a*b*c**2*atan(c*x**3)/3 - a*b*c/(3*x**3) - a*b *atan(c*x**3)/(3*x**6) + b**2*c**3*sqrt(-1/c**2)*atan(c*x**3)/3 + b**2*c** 2*log(x) - b**2*c**2*log(x - (-1/c**2)**(1/6))/3 - b**2*c**2*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/3 - b**2*c**2*atan(c*x**3)**2/6 - b**2*c*atan(c*x**3)/(3*x**3) - b**2*atan(c*x**3)**2/(6*x**6), Ne(c, 0)) , (-a**2/(6*x**6), True))
Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {1}{3} \, {\left ({\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c + \frac {\arctan \left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{6} \, {\left ({\left (\arctan \left (c x^{3}\right )^{2} - \log \left (c^{2} x^{6} + 1\right ) + 6 \, \log \left (x\right )\right )} c^{2} - 2 \, {\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c \arctan \left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \arctan \left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \]
-1/3*((c*arctan(c*x^3) + 1/x^3)*c + arctan(c*x^3)/x^6)*a*b + 1/6*((arctan( c*x^3)^2 - log(c^2*x^6 + 1) + 6*log(x))*c^2 - 2*(c*arctan(c*x^3) + 1/x^3)* c*arctan(c*x^3))*b^2 - 1/6*b^2*arctan(c*x^3)^2/x^6 - 1/6*a^2/x^6
\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{7}} \,d x } \]
Time = 0.72 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=b^2\,c^2\,\ln \left (x\right )-\frac {b^2\,c^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6}-\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c^2\,x^6+1\right )}{6}-\frac {a^2}{6\,x^6}-\frac {b^2\,c\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^3}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,c^2\,\mathrm {atan}\left (\frac {a^2\,c\,x^3}{a^2+49\,b^2}+\frac {49\,b^2\,c\,x^3}{a^2+49\,b^2}\right )}{3}-\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^6} \]