Integrand size = 24, antiderivative size = 142 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \]
1/a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-1/a^3/c^2/arctan(a*x)/(a^2*c*x^2+c )^(1/2)-1/4*Si(arctan(a*x))*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)+ 3/4*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)
Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {4 a^2 x^2+\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \text {Si}(\arctan (a x))-3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)} \]
-1/4*(4*a^2*x^2 + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*SinIntegral[ArcTan[a*x]] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*SinIntegral[3*ArcTan[a*x]])/(a^3*c^2* (1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])
Time = 1.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5499, 5437, 5506, 5505, 3042, 3780, 4906, 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.
| \(\displaystyle \int \frac {x^2}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle \frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5506 |
| \(\displaystyle \frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \arctan (a x)}+\frac {\sin (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
(-(1/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (Sqrt[1 + a^2*x^2]*SinIntegr al[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2]))/(a^2*c) - (-(1/(a*c*(c + a^2*c *x^2)^(3/2)*ArcTan[a*x])) - (3*Sqrt[1 + a^2*x^2]*(SinIntegral[ArcTan[a*x]] /4 + SinIntegral[3*ArcTan[a*x]]/4))/(a*c^2*Sqrt[c + a^2*c*x^2]))/a^2
3.6.90.3.1 Defintions of rubi rules used
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_)]*(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_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e 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] && IGtQ[m, 1] && 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])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !(I ntegerQ[q] || GtQ[d, 0])
Result contains complex when optimal does not.
Time = 13.75 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.18
| method | result | size |
| default | \(-\frac {i \left (\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}-3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}-6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-8 i \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-3 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+3 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) a^{3} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(309\) |
-1/8*I*(arctan(a*x)*Ei(1,-I*arctan(a*x))*a^4*x^4-3*arctan(a*x)*Ei(1,-3*I*a rctan(a*x))*a^4*x^4-arctan(a*x)*Ei(1,I*arctan(a*x))*a^4*x^4+3*arctan(a*x)* Ei(1,3*I*arctan(a*x))*a^4*x^4+2*arctan(a*x)*Ei(1,-I*arctan(a*x))*a^2*x^2-6 *arctan(a*x)*Ei(1,-3*I*arctan(a*x))*a^2*x^2-2*arctan(a*x)*Ei(1,I*arctan(a* x))*a^2*x^2+6*arctan(a*x)*Ei(1,3*I*arctan(a*x))*a^2*x^2-8*I*(a^2*x^2+1)^(1 /2)*a^2*x^2+Ei(1,-I*arctan(a*x))*arctan(a*x)-3*Ei(1,-3*I*arctan(a*x))*arct an(a*x)-Ei(1,I*arctan(a*x))*arctan(a*x)+3*Ei(1,3*I*arctan(a*x))*arctan(a*x ))/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/arctan(a*x)/a^3/c^3/(a^4*x^ 4+2*a^2*x^2+1)
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
integral(sqrt(a^2*c*x^2 + c)*x^2/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3 *x^2 + c^3)*arctan(a*x)^2), x)
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]