Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {1}{a c^3 x^2 \arctan (a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {2 a \text {Si}(2 \arctan (a x))}{c^3}+\frac {a \text {Si}(4 \arctan (a x))}{2 c^3}-\frac {2 \text {Int}\left (\frac {1}{x^3 \arctan (a x)},x\right )}{a c^3} \]
-1/a/c^3/x^2/arctan(a*x)+a/c^3/(a^2*x^2+1)^2/arctan(a*x)+a/c^3/(a^2*x^2+1) /arctan(a*x)+2*a*Si(2*arctan(a*x))/c^3+1/2*a*Si(4*arctan(a*x))/c^3-2*Unint egrable(1/x^3/arctan(a*x),x)/a/c^3
Not integrable
Time = 1.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx \]
Not integrable
Time = 1.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 27, 5437, 5501, 5437, 5461, 5377, 5505, 4906, 27, 2009, 3042, 3780}
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.
\(\displaystyle \int \frac {1}{x^2 \arctan (a x)^2 \left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\int \frac {1}{c^2 x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{c}-a^2 \int \frac {1}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{c^3}-\frac {a^2 \int \frac {1}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx}{c^3}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{c^3}-\frac {a^2 \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}dx-a^2 \int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{c^3}-\frac {a^2 \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}dx-a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^3}-\frac {a^2 \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 5461 |
\(\displaystyle \frac {-\left (a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 5377 |
\(\displaystyle \frac {-\left (a^2 \left (-2 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {-\left (a^2 \left (-\frac {2 \int \frac {a x}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {4 \int \frac {a x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {-\left (a^2 \left (-\frac {2 \int \frac {\sin (2 \arctan (a x))}{2 \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {4 \int \left (\frac {\sin (2 \arctan (a x))}{4 \arctan (a x)}+\frac {\sin (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\left (a^2 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {4 \int \left (\frac {\sin (2 \arctan (a x))}{4 \arctan (a x)}+\frac {\sin (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}\right )}{c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\left (a^2 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}\right )}{c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\left (a^2 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )\right )-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}\right )}{c^3}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {-\frac {2 \int \frac {1}{x^3 \arctan (a x)}dx}{a}-\left (a^2 \left (-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}-\frac {\text {Si}(2 \arctan (a x))}{a}\right )\right )-\frac {1}{a x^2 \arctan (a x)}}{c^3}-\frac {a^2 \left (-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}\right )}{c^3}\) |
3.6.64.3.1 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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
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_)*((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[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 = 10.73 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{2}}d x\]
Not integrable
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 1.82 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{6} x^{8} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + x^{2} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
Integral(1/(a**6*x**8*atan(a*x)**2 + 3*a**4*x**6*atan(a*x)**2 + 3*a**2*x** 4*atan(a*x)**2 + x**2*atan(a*x)**2), x)/c**3
Not integrable
Time = 0.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 6.36 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2}} \,d x } \]
-((a^5*c^3*x^6 + 2*a^3*c^3*x^4 + a*c^3*x^2)*arctan(a*x)*integrate(2*(3*a^2 *x^2 + 1)/((a^7*c^3*x^9 + 3*a^5*c^3*x^7 + 3*a^3*c^3*x^5 + a*c^3*x^3)*arcta n(a*x)), x) + 1)/((a^5*c^3*x^6 + 2*a^3*c^3*x^4 + a*c^3*x^2)*arctan(a*x))
Not integrable
Time = 128.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]