Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^2 x^2 \arctan (a x)^2}+\frac {a}{2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {a^2 x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {a \operatorname {CosIntegral}(2 \arctan (a x))}{c^2}-\frac {\text {Int}\left (\frac {1}{x^3 \arctan (a x)^2},x\right )}{a c^2} \]
-1/2/a/c^2/x^2/arctan(a*x)^2+1/2*a/c^2/(a^2*x^2+1)/arctan(a*x)^2-a^2*x/c^2 /(a^2*x^2+1)/arctan(a*x)+a*Ci(2*arctan(a*x))/c^2-Unintegrable(1/x^3/arctan (a*x)^2,x)/a/c^2
Not integrable
Time = 2.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx \]
Not integrable
Time = 1.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 27, 5437, 5461, 5377, 5503, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 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.
\(\displaystyle \int \frac {1}{x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\int \frac {1}{c x^2 \left (a^2 x^2+1\right ) \arctan (a x)^3}dx}{c}-a^2 \int \frac {1}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^3}dx}{c^2}-\frac {a^2 \int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^2}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^3}dx}{c^2}-\frac {a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 5461 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 5377 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a}-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 5439 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {1}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-\frac {\int \frac {a^2 x^2}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-\frac {\int \frac {\sin (\arctan (a x))^2}{\arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-\frac {\int \left (\frac {1}{2 \arctan (a x)}-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\int \frac {1}{x^3 \arctan (a x)^2}dx}{a}-\frac {1}{2 a x^2 \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-a \left (-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )-\frac {1}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}\right )}{c^2}\) |
3.7.32.3.1 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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Sy mbol] :> Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c , d, m, n, p}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*Arc Tan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_))/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*( p + 1))), x] - Simp[f*(m/(b*c*d*(p + 1))) Int[(f*x)^(m - 1)*(a + b*ArcTan [c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & & LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sin[x]^m/ Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p }, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q ] || GtQ[d, 0])
Not integrable
Time = 9.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}d x\]
Not integrable
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]
Not integrable
Time = 1.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + x^{2} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
Not integrable
Time = 0.35 (sec) , antiderivative size = 142, normalized size of antiderivative = 6.45 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]
1/2*(2*(a^4*c^2*x^5 + a^2*c^2*x^3)*arctan(a*x)^2*integrate((6*a^4*x^4 + 7* a^2*x^2 + 3)/((a^6*c^2*x^8 + 2*a^4*c^2*x^6 + a^2*c^2*x^4)*arctan(a*x)), x) - a*x + 2*(2*a^2*x^2 + 1)*arctan(a*x))/((a^4*c^2*x^5 + a^2*c^2*x^3)*arcta n(a*x)^2)
Not integrable
Time = 128.17 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]
Not integrable
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]