\(\int \frac {x^5 \arctan (a x)^2}{(c+a^2 c x^2)^{5/2}} \, dx\) [347]

Optimal result

Integrand size = 24, antiderivative size = 400 \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2}{27 a^6 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {32}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 x \arctan (a x)}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^6 c^3}+\frac {4 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

2/27/a^6/c/(a^2*c*x^2+c)^(3/2)-2/9*x^3*arctan(a*x)/a^3/c/(a^2*c*x^2+c)^(3/2)+1/3*x^2*arctan(a*x)^2/a^4/c/(a^2*
c*x^2+c)^(3/2)-32/9/a^6/c^2/(a^2*c*x^2+c)^(1/2)-10/3*x*arctan(a*x)/a^5/c^2/(a^2*c*x^2+c)^(1/2)+5/3*arctan(a*x)
^2/a^6/c^2/(a^2*c*x^2+c)^(1/2)+4*I*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c
^2/(a^2*c*x^2+c)^(1/2)-2*I*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+
c)^(1/2)+2*I*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+arctan
(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^6/c^3

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5084, 5050, 5010, 5006, 5014, 5060, 272, 45} \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^6 c^3}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {4 i \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {32}{9 a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {2}{27 a^6 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {10 x \arctan (a x)}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

Int[(x^5*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]

[Out]

2/(27*a^6*c*(c + a^2*c*x^2)^(3/2)) - 32/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*a^3*c*(c + a^
2*c*x^2)^(3/2)) - (10*x*ArcTan[a*x])/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a^4*c*(c + a^2*c
*x^2)^(3/2)) + (5*ArcTan[a*x]^2)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^6*c^
3) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2
]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]
) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2])

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 5014

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rule 5050

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] - Dist[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]

Rule 5060

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*ArcTan[c*x])^(p - 1)/(c*d*m^2)), x] + (Dist[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] - Dist[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] - Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p
/(c^2*d*m)), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && G
tQ[p, 1]

Rule 5084

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int[
x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*Arc
Tan[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c} \\ & = -\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a^2}+\frac {\int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {2 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac {\int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c} \\ & = -\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^6 c^3}+\frac {\text {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{9 a^2}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac {4 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^5 c}-\frac {2 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c} \\ & = -\frac {10}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 x \arctan (a x)}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^6 c^3}+\frac {\text {Subst}\left (\int \left (-\frac {1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac {1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{9 a^2}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2}{27 a^6 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {32}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 x \arctan (a x)}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^6 c^3}+\frac {4 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.57 \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {8 (-95+\cos (2 \arctan (a x)))-9 \left (1+a^2 x^2\right ) \arctan (a x)^2 (-45-20 \cos (2 \arctan (a x))+\cos (4 \arctan (a x)))-432 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+432 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+6 \arctan (a x) \left (-124 a x-72 \sqrt {1+a^2 x^2} \log \left (1-i e^{i \arctan (a x)}\right )+72 \sqrt {1+a^2 x^2} \log \left (1+i e^{i \arctan (a x)}\right )+\left (1+a^2 x^2\right ) \sin (4 \arctan (a x))\right )}{216 a^6 c^2 \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[(x^5*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]

[Out]

(8*(-95 + Cos[2*ArcTan[a*x]]) - 9*(1 + a^2*x^2)*ArcTan[a*x]^2*(-45 - 20*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]
]) - (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, I*E^(
I*ArcTan[a*x])] + 6*ArcTan[a*x]*(-124*a*x - 72*Sqrt[1 + a^2*x^2]*Log[1 - I*E^(I*ArcTan[a*x])] + 72*Sqrt[1 + a^
2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] + (1 + a^2*x^2)*Sin[4*ArcTan[a*x]]))/(216*a^6*c^2*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{6} c^{3}}+\frac {7 \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{6} \left (a^{2} x^{2}+1\right )}-\frac {7 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 c^{3} a^{6} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right )}{216 c^{3} a^{6} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{6} c^{3}}+\frac {2 \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3} a^{6}}\)	\(454\)

[In]

int(x^5*arctan(a*x)^2/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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^6/c^3+7/8*(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^6/(a^2*x^2+1)-7/8*
(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^6/(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^6/(a^4*x^4+2*a^2*x^2+1)
+arctan(a*x)^2*(c*(a*x-I)*(I+a*x))^(1/2)/a^6/c^3+2*(arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x
)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*a*x)/(a^2*x^2+
1)^(1/2)))*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/c^3/a^6

Fricas [F]

\[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{5} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^5*arctan(a*x)^2/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^5*arctan(a*x)^2/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3), x)

Sympy [F]

\[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**5*atan(a*x)**2/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x**5*atan(a*x)**2/(c*(a**2*x**2 + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{5} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^5*arctan(a*x)^2/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^5*arctan(a*x)^2/(a^2*c*x^2 + c)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*arctan(a*x)^2/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int((x^5*atan(a*x)^2)/(c + a^2*c*x^2)^(5/2),x)

[Out]

int((x^5*atan(a*x)^2)/(c + a^2*c*x^2)^(5/2), x)