Integrand size = 22, antiderivative size = 199 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 x}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x}{27 a c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^2}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}} \]
-2/27*x/a/c/(a^2*c*x^2+c)^(3/2)+2/9*arctan(a*x)/a^2/c/(a^2*c*x^2+c)^(3/2)+ 1/3*x*arctan(a*x)^2/a/c/(a^2*c*x^2+c)^(3/2)-1/3*arctan(a*x)^3/a^2/c/(a^2*c *x^2+c)^(3/2)-40/27*x/a/c^2/(a^2*c*x^2+c)^(1/2)+4/3*arctan(a*x)/a^2/c^2/(a ^2*c*x^2+c)^(1/2)+2/3*x*arctan(a*x)^2/a/c^2/(a^2*c*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.46 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-2 a x \left (21+20 a^2 x^2\right )+6 \left (7+6 a^2 x^2\right ) \arctan (a x)+9 a x \left (3+2 a^2 x^2\right ) \arctan (a x)^2-9 \arctan (a x)^3\right )}{27 c^3 \left (a+a^3 x^2\right )^2} \]
(Sqrt[c + a^2*c*x^2]*(-2*a*x*(21 + 20*a^2*x^2) + 6*(7 + 6*a^2*x^2)*ArcTan[ a*x] + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^2 - 9*ArcTan[a*x]^3))/(27*c^3*(a + a^3*x^2)^2)
Time = 0.53 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5465, 5435, 209, 208, 5433, 208}
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.
| \(\displaystyle \int \frac {x \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{5/2}}dx}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {\frac {2 \int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}-\frac {2}{9} \int \frac {1}{\left (a^2 c x^2+c\right )^{5/2}}dx+\frac {x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {2 \int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}-\frac {2}{9} \left (\frac {2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {2 \int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2}{9} \left (\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
| \(\Big \downarrow \) 5433 |
| \(\displaystyle \frac {\frac {2 \left (-2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2}{9} \left (\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}\right )}{3 c}-\frac {2}{9} \left (\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
-1/3*ArcTan[a*x]^3/(a^2*c*(c + a^2*c*x^2)^(3/2)) + ((-2*(x/(3*c*(c + a^2*c *x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c + a^2*c*x^2])))/9 + (2*ArcTan[a*x])/(9* a*c*(c + a^2*c*x^2)^(3/2)) + (x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*((-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2* c*x^2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2])))/(3*c))/a
3.5.54.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p - 1) Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ {a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{2} c^{3}}-\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right )}{216 c^{3} a^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(312\) |
1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(I*a^3*x^3+3*a ^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^2/a^2/c^3-1/8*(arc tan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x-I)*(I+a* x))^(1/2)/c^3/a^2/(a^2*x^2+1)+1/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arc tan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)/c^3/a^2/(a^2*x^2+1)-1/216* (c*(a*x-I)*(I+a*x))^(1/2)*(I*a^3*x^3-3*a^2*x^2-3*I*a*x+1)*(-9*I*arctan(a*x )^2+9*arctan(a*x)^3+2*I-6*arctan(a*x))/c^3/a^2/(a^4*x^4+2*a^2*x^2+1)
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.52 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (40 \, a^{3} x^{3} - 9 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 9 \, \arctan \left (a x\right )^{3} + 42 \, a x - 6 \, {\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{27 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
-1/27*(40*a^3*x^3 - 9*(2*a^3*x^3 + 3*a*x)*arctan(a*x)^2 + 9*arctan(a*x)^3 + 42*a*x - 6*(6*a^2*x^2 + 7)*arctan(a*x))*sqrt(a^2*c*x^2 + c)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)
\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]