Integrand size = 24, antiderivative size = 172 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2}{27 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {14}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 x \arctan (a x)}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^2}{3 a^4 c^2 \sqrt {c+a^2 c x^2}} \]
-2/27/a^4/c/(a^2*c*x^2+c)^(3/2)+2/9*x^3*arctan(a*x)/a/c/(a^2*c*x^2+c)^(3/2 )-1/3*x^2*arctan(a*x)^2/a^2/c/(a^2*c*x^2+c)^(3/2)+14/9/a^4/c^2/(a^2*c*x^2+ c)^(1/2)+4/3*x*arctan(a*x)/a^3/c^2/(a^2*c*x^2+c)^(1/2)-2/3*arctan(a*x)^2/a ^4/c^2/(a^2*c*x^2+c)^(1/2)
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.47 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (40+42 a^2 x^2+6 a x \left (6+7 a^2 x^2\right ) \arctan (a x)-9 \left (2+3 a^2 x^2\right ) \arctan (a x)^2\right )}{27 a^4 c^3 \left (1+a^2 x^2\right )^2} \]
(Sqrt[c + a^2*c*x^2]*(40 + 42*a^2*x^2 + 6*a*x*(6 + 7*a^2*x^2)*ArcTan[a*x] - 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^2))/(27*a^4*c^3*(1 + a^2*x^2)^2)
Time = 0.69 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5475, 243, 53, 2009, 5465, 5429}
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.
\(\displaystyle \int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5475 |
\(\displaystyle \frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {2}{9} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2}}dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \left (\frac {1}{a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {1}{a^2 \left (a^2 c x^2+c\right )^{5/2}}\right )dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 5429 |
\(\displaystyle -\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
(-2/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + 2/(a^4*c^2*Sqrt[c + a^2*c*x^2]))/9 + (2*x^3*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x]^2)/( 3*a^2*c*(c + a^2*c*x^2)^(3/2)) + (2*(-(ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^2*c *x^2])) + (2*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^ 2*c*x^2])))/a))/(3*a^2*c)
3.4.49.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_.) *(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Ar cTan[c*x])^(p - 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + Simp[f^2*((m - 1)/(c^2*d*m)) Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[ b^2*p*((p - 1)/m^2) Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2* q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
Result contains complex when optimal does not.
Time = 1.87 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{4} c^{3}}-\frac {3 \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{4} \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 c^{3} a^{4} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right )}{216 c^{3} a^{4} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(276\) |
-1/216*(6*I*arctan(a*x)+9*arctan(a*x)^2-2)*(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^4/c^3-3/8*(arctan(a*x)^2-2+2*I* arctan(a*x))*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/c^3/a^4/(a^2*x^2+1)+3/8*( c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)^2-2-2*I*arctan(a*x))/c^3/a ^4/(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)*(-6*I*arctan(a*x)+9*arctan(a*x)^2-2)/c^3/a^4/(a^4*x^4+2*a^2*x^2+1)
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.53 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (42 \, a^{2} x^{2} - 9 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )^{2} + 6 \, {\left (7 \, a^{3} x^{3} + 6 \, a x\right )} \arctan \left (a x\right ) + 40\right )}}{27 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
1/27*sqrt(a^2*c*x^2 + c)*(42*a^2*x^2 - 9*(3*a^2*x^2 + 2)*arctan(a*x)^2 + 6 *(7*a^3*x^3 + 6*a*x)*arctan(a*x) + 40)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4* c^3)
\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]