Integrand size = 22, antiderivative size = 170 \[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^6 c^3}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^6 c^{5/2}} \]
-1/9*x^3/a^3/c/(a^2*c*x^2+c)^(3/2)+1/3*x^2*arctan(a*x)/a^4/c/(a^2*c*x^2+c) ^(3/2)-arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^6/c^(5/2)-5/3*x/a^5/c^2/ (a^2*c*x^2+c)^(1/2)+5/3*arctan(a*x)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+arctan(a*x )*(a^2*c*x^2+c)^(1/2)/a^6/c^3
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.77 \[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a x \left (15+16 a^2 x^2\right ) \sqrt {c+a^2 c x^2}-3 \sqrt {c+a^2 c x^2} \left (8+12 a^2 x^2+3 a^4 x^4\right ) \arctan (a x)+9 \sqrt {c} \left (1+a^2 x^2\right )^2 \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{9 a^6 c^3 \left (1+a^2 x^2\right )^2} \]
-1/9*(a*x*(15 + 16*a^2*x^2)*Sqrt[c + a^2*c*x^2] - 3*Sqrt[c + a^2*c*x^2]*(8 + 12*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + 9*Sqrt[c]*(1 + a^2*x^2)^2*Log[a*c *x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^6*c^3*(1 + a^2*x^2)^2)
Time = 1.06 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.48, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5499, 5473, 5465, 208, 5499, 5465, 208, 224, 219}
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.
| \(\displaystyle \int \frac {x^5 \arctan (a x)}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{5/2}}dx}{a^2}\) |
| \(\Big \downarrow \) 5473 |
| \(\displaystyle \frac {\int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \int \frac {x \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a^2 c}-\frac {\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
-((x^3/(9*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x])/(3*a^2*c*(c + a^2 *c*x^2)^(3/2)) + (2*(x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt [c + a^2*c*x^2])))/(3*a^2*c))/a^2) + (-((x/(a*c*Sqrt[c + a^2*c*x^2]) - Arc Tan[a*x]/(a^2*c*Sqrt[c + a^2*c*x^2]))/a^2) + ((Sqrt[c + a^2*c*x^2]*ArcTan[ a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c]))/ (a^2*c))/(a^2*c)
3.3.40.3.1 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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[b*(f*x)^m*((d + e*x^2)^(q + 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(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]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2 *d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -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]
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.92
| method | result | size |
| default | \(\frac {\left (9 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) a^{4} x^{4}-9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) a^{4} x^{4}-16 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+36 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+18 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) a^{2} x^{2}-18 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) a^{2} x^{2}-15 \sqrt {a^{2} x^{2}+1}\, a x +24 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}+9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )-9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{9 \sqrt {a^{2} x^{2}+1}\, a^{6} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(327\) |
1/9*(9*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^4*x^4+9*ln((1+I*a*x)/(a^2*x^2+1)^(1 /2)-I)*a^4*x^4-9*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)*a^4*x^4-16*(a^2*x^2+1)^ (1/2)*a^3*x^3+36*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2+18*ln((1+I*a*x)/(a^ 2*x^2+1)^(1/2)-I)*a^2*x^2-18*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)*a^2*x^2-15* (a^2*x^2+1)^(1/2)*a*x+24*arctan(a*x)*(a^2*x^2+1)^(1/2)+9*ln((1+I*a*x)/(a^2 *x^2+1)^(1/2)-I)-9*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I))/(a^2*x^2+1)^(1/2)*(c *(a*x-I)*(I+a*x))^(1/2)/a^6/c^3/(a^4*x^4+2*a^2*x^2+1)
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {9 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (16 \, a^{3} x^{3} + 15 \, a x - 3 \, {\left (3 \, a^{4} x^{4} + 12 \, a^{2} x^{2} + 8\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{18 \, {\left (a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}\right )}} \]
1/18*(9*(a^4*x^4 + 2*a^2*x^2 + 1)*sqrt(c)*log(-2*a^2*c*x^2 + 2*sqrt(a^2*c* x^2 + c)*a*sqrt(c)*x - c) - 2*(16*a^3*x^3 + 15*a*x - 3*(3*a^4*x^4 + 12*a^2 *x^2 + 8)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/(a^10*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3)
\[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{5} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]