Integrand size = 20, antiderivative size = 88 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a}{6 c x^2}-\frac {\arctan (a x)}{3 c x^3}+\frac {a^2 \arctan (a x)}{c x}+\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c} \]
-1/6*a/c/x^2-1/3*arctan(a*x)/c/x^3+a^2*arctan(a*x)/c/x+1/2*a^3*arctan(a*x) ^2/c-4/3*a^3*ln(x)/c+2/3*a^3*ln(a^2*x^2+1)/c
Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a}{6 c x^2}-\frac {\arctan (a x)}{3 c x^3}+\frac {a^2 \arctan (a x)}{c x}+\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c} \]
-1/6*a/(c*x^2) - ArcTan[a*x]/(3*c*x^3) + (a^2*ArcTan[a*x])/(c*x) + (a^3*Ar cTan[a*x]^2)/(2*c) - (4*a^3*Log[x])/(3*c) + (2*a^3*Log[1 + a^2*x^2])/(3*c)
Time = 0.62 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5453, 27, 5361, 243, 54, 2009, 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.
| \(\displaystyle \int \frac {\arctan (a x)}{x^4 \left (a^2 c x^2+c\right )} \, dx\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4}dx}{c}-a^2 \int \frac {\arctan (a x)}{c x^2 \left (a^2 x^2+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4}dx}{c}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {1}{3} a \int \frac {1}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {1}{6} a \int \left (\frac {a^4}{a^2 x^2+1}-\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )}{c}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )}{c}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c}-\frac {a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )}{c}\) |
-((a^2*(-(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^ 2*x^2]))/2))/c) + (-1/3*ArcTan[a*x]/x^3 + (a*(-x^(-2) - a^2*Log[x^2] + a^2 *Log[1 + a^2*x^2]))/6)/c
3.2.81.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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_)]*(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 = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92
| method | result | size |
| parallelrisch | \(-\frac {-3 a^{3} \arctan \left (a x \right )^{2} x^{3}+8 \ln \left (x \right ) a^{3} x^{3}-4 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-2 a^{3} x^{3}-6 a^{2} \arctan \left (a x \right ) x^{2}+a x +2 \arctan \left (a x \right )}{6 c \,x^{3}}\) | \(81\) |
| derivativedivides | \(a^{3} \left (\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )}{c a x}-\frac {-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )+\frac {3 \arctan \left (a x \right )^{2}}{2}}{3 c}\right )\) | \(85\) |
| default | \(a^{3} \left (\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )}{c a x}-\frac {-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )+\frac {3 \arctan \left (a x \right )^{2}}{2}}{3 c}\right )\) | \(85\) |
| parts | \(\frac {a^{3} \arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )}{c x}-\frac {a^{3} \left (-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )\right )+\frac {3 a^{3} \arctan \left (a x \right )^{2}}{2}}{3 c}\) | \(89\) |
| risch | \(-\frac {a^{3} \ln \left (i a x +1\right )^{2}}{8 c}+\frac {\left (3 a^{3} x^{3} \ln \left (-i a x +1\right )-6 i a^{2} x^{2}+2 i\right ) \ln \left (i a x +1\right )}{12 c \,x^{3}}-\frac {3 a^{3} \ln \left (-i a x +1\right )^{2} x^{3}+32 \ln \left (x \right ) a^{3} x^{3}-16 \ln \left (3 a^{2} x^{2}+3\right ) a^{3} x^{3}-12 i a^{2} x^{2} \ln \left (-i a x +1\right )+4 i \ln \left (-i a x +1\right )+4 a x}{24 c \,x^{3}}\) | \(152\) |
-1/6*(-3*a^3*arctan(a*x)^2*x^3+8*ln(x)*a^3*x^3-4*a^3*ln(a^2*x^2+1)*x^3-2*a ^3*x^3-6*a^2*arctan(a*x)*x^2+a*x+2*arctan(a*x))/c/x^3
Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {3 \, a^{3} x^{3} \arctan \left (a x\right )^{2} + 4 \, a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{3} x^{3} \log \left (x\right ) - a x + 2 \, {\left (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{6 \, c x^{3}} \]
1/6*(3*a^3*x^3*arctan(a*x)^2 + 4*a^3*x^3*log(a^2*x^2 + 1) - 8*a^3*x^3*log( x) - a*x + 2*(3*a^2*x^2 - 1)*arctan(a*x))/(c*x^3)
Time = 0.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\begin {cases} - \frac {4 a^{3} \log {\left (x \right )}}{3 c} + \frac {2 a^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 c} + \frac {a^{3} \operatorname {atan}^{2}{\left (a x \right )}}{2 c} + \frac {a^{2} \operatorname {atan}{\left (a x \right )}}{c x} - \frac {a}{6 c x^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{3 c x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-4*a**3*log(x)/(3*c) + 2*a**3*log(x**2 + a**(-2))/(3*c) + a**3* atan(a*x)**2/(2*c) + a**2*atan(a*x)/(c*x) - a/(6*c*x**2) - atan(a*x)/(3*c* x**3), Ne(a, 0)), (0, True))
Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {1}{3} \, {\left (\frac {3 \, a^{3} \arctan \left (a x\right )}{c} + \frac {3 \, a^{2} x^{2} - 1}{c x^{3}}\right )} \arctan \left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \left (x\right ) + 1\right )} a}{6 \, c x^{2}} \]
1/3*(3*a^3*arctan(a*x)/c + (3*a^2*x^2 - 1)/(c*x^3))*arctan(a*x) - 1/6*(3*a ^2*x^2*arctan(a*x)^2 - 4*a^2*x^2*log(a^2*x^2 + 1) + 8*a^2*x^2*log(x) + 1)* a/(c*x^2)
\[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
Time = 0.55 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {2\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c}-\frac {\mathrm {atan}\left (a\,x\right )}{3\,c\,x^3}-\frac {a}{6\,c\,x^2}-\frac {4\,a^3\,\ln \left (x\right )}{3\,c}+\frac {a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,c}+\frac {a^2\,\mathrm {atan}\left (a\,x\right )}{c\,x} \]