Integrand size = 18, antiderivative size = 63 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=-\frac {a c}{6 x^2}-\frac {c \arctan (a x)}{3 x^3}-\frac {a^2 c \arctan (a x)}{x}+\frac {2}{3} a^3 c \log (x)-\frac {1}{3} a^3 c \log \left (1+a^2 x^2\right ) \]
-1/6*a*c/x^2-1/3*c*arctan(a*x)/x^3-a^2*c*arctan(a*x)/x+2/3*a^3*c*ln(x)-1/3 *a^3*c*ln(a^2*x^2+1)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=\frac {c \left (-2 \left (1+3 a^2 x^2\right ) \arctan (a x)+a x \left (-1+4 a^2 x^2 \log (x)-2 a^2 x^2 \log \left (1+a^2 x^2\right )\right )\right )}{6 x^3} \]
(c*(-2*(1 + 3*a^2*x^2)*ArcTan[a*x] + a*x*(-1 + 4*a^2*x^2*Log[x] - 2*a^2*x^ 2*Log[1 + a^2*x^2])))/(6*x^3)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5485, 5361, 243, 47, 14, 16, 54, 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.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int \frac {\arctan (a x)}{x^2}dx+c \int \frac {\arctan (a x)}{x^4}dx\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle a^2 c \left (a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )+c \left (\frac {1}{3} a \int \frac {1}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{3 x^3}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle a^2 c \left (\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )+c \left (\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}\right )\) |
\(\Big \downarrow \) 47 |
\(\displaystyle a^2 c \left (\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )+c \left (\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}\right )\) |
\(\Big \downarrow \) 14 |
\(\displaystyle a^2 c \left (\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )+c \left (\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle c \left (\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}\right )+a^2 c \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle c \left (\frac {1}{6} a \int \left (\frac {a^4}{a^2 x^2+1}-\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\arctan (a x)}{3 x^3}\right )+a^2 c \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 c \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )+c \left (\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}\right )\) |
a^2*c*(-(ArcTan[a*x]/x) + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) + c*(-1/3*A rcTan[a*x]/x^3 + (a*(-x^(-2) - a^2*Log[x^2] + a^2*Log[1 + a^2*x^2]))/6)
3.2.56.3.1 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[((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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89
method | result | size |
parts | \(-\frac {a^{2} c \arctan \left (a x \right )}{x}-\frac {c \arctan \left (a x \right )}{3 x^{3}}-\frac {c a \left (a^{2} \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 x^{2}}-2 a^{2} \ln \left (x \right )\right )}{3}\) | \(56\) |
derivativedivides | \(a^{3} \left (-\frac {c \arctan \left (a x \right )}{a x}-\frac {c \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c \left (\ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-2 \ln \left (a x \right )\right )}{3}\right )\) | \(60\) |
default | \(a^{3} \left (-\frac {c \arctan \left (a x \right )}{a x}-\frac {c \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c \left (\ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-2 \ln \left (a x \right )\right )}{3}\right )\) | \(60\) |
parallelrisch | \(\frac {4 c \,a^{3} \ln \left (x \right ) x^{3}-2 c \,a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}+a^{3} c \,x^{3}-6 a^{2} c \,x^{2} \arctan \left (a x \right )-a c x -2 c \arctan \left (a x \right )}{6 x^{3}}\) | \(70\) |
risch | \(\frac {i c \left (3 a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )}{6 x^{3}}+\frac {c \left (4 \ln \left (x \right ) a^{3} x^{3}-2 \ln \left (-3 a^{2} x^{2}-3\right ) a^{3} x^{3}-3 i a^{2} x^{2} \ln \left (-i a x +1\right )-a x -i \ln \left (-i a x +1\right )\right )}{6 x^{3}}\) | \(95\) |
meijerg | \(\frac {a^{3} c \left (4 \ln \left (x \right )+4 \ln \left (a \right )-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {a^{3} c \left (-\frac {2}{a^{2} x^{2}}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (a \right )}{3}+\frac {-\frac {4 a^{2} x^{2}}{9}+\frac {4}{3}}{a^{2} x^{2}}-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4}\) | \(131\) |
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=-\frac {2 \, a^{3} c x^{3} \log \left (a^{2} x^{2} + 1\right ) - 4 \, a^{3} c x^{3} \log \left (x\right ) + a c x + 2 \, {\left (3 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{6 \, x^{3}} \]
-1/6*(2*a^3*c*x^3*log(a^2*x^2 + 1) - 4*a^3*c*x^3*log(x) + a*c*x + 2*(3*a^2 *c*x^2 + c)*arctan(a*x))/x^3
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=\begin {cases} \frac {2 a^{3} c \log {\left (x \right )}}{3} - \frac {a^{3} c \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3} - \frac {a^{2} c \operatorname {atan}{\left (a x \right )}}{x} - \frac {a c}{6 x^{2}} - \frac {c \operatorname {atan}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((2*a**3*c*log(x)/3 - a**3*c*log(x**2 + a**(-2))/3 - a**2*c*atan( a*x)/x - a*c/(6*x**2) - c*atan(a*x)/(3*x**3), Ne(a, 0)), (0, True))
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=-\frac {1}{6} \, {\left (2 \, a^{2} c \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} c \log \left (x^{2}\right ) + \frac {c}{x^{2}}\right )} a - \frac {{\left (3 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{3 \, x^{3}} \]
-1/6*(2*a^2*c*log(a^2*x^2 + 1) - 2*a^2*c*log(x^2) + c/x^2)*a - 1/3*(3*a^2* c*x^2 + c)*arctan(a*x)/x^3
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{x^{4}} \,d x } \]
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^4} \, dx=\frac {c\,\left (4\,a^3\,\ln \left (x\right )-2\,a^3\,\ln \left (a^2\,x^2+1\right )\right )}{6}-\frac {\frac {c\,\mathrm {atan}\left (a\,x\right )}{3}+\frac {a\,c\,x}{6}+a^2\,c\,x^2\,\mathrm {atan}\left (a\,x\right )}{x^3} \]