\(\int x^3 (c+a^2 c x^2)^{5/2} \arctan (a x)^2 \, dx\) [323]

Optimal result

Integrand size = 24, antiderivative size = 578 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=-\frac {115 c^2 \sqrt {c+a^2 c x^2}}{4032 a^4}-\frac {115 c \left (c+a^2 c x^2\right )^{3/2}}{18144 a^4}-\frac {23 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac {\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac {47 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{1344 a^3}-\frac {205 c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{6048 a}-\frac {103 a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{63 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{63 a^2}+\frac {5}{21} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {19}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {115 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2016 a^4 \sqrt {c+a^2 c x^2}}+\frac {115 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {c+a^2 c x^2}} \]

[Out]

-115/18144*c*(a^2*c*x^2+c)^(3/2)/a^4-23/7560*(a^2*c*x^2+c)^(5/2)/a^4+1/252*(a^2*c*x^2+c)^(7/2)/a^4/c+115/4032*
I*c^3*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/2016*I*c^3*a
rctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/4032*I*c^3*po
lylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/4032*c^2*(a^2*c*x^2+c
)^(1/2)/a^4+47/1344*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^3-205/6048*c^2*x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)
/a-103/1512*a*c^2*x^5*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/36*a^3*c^2*x^7*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-2/63*c^
2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^4+1/63*c^2*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+5/21*c^2*x^4*arctan
(a*x)^2*(a^2*c*x^2+c)^(1/2)+19/63*a^2*c^2*x^6*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/9*a^4*c^2*x^8*arctan(a*x)^2*
(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 7.39 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.00, number of steps used = 203, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5070, 5072, 267, 5010, 5006, 5050, 272, 45} \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {c^2 x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{63 a^2}+\frac {19}{63} a^2 c^2 x^6 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {103 a c^2 x^5 \arctan (a x) \sqrt {a^2 c x^2+c}}{1512}+\frac {5}{21} c^2 x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {205 c^2 x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{6048 a}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2016 a^4 \sqrt {a^2 c x^2+c}}-\frac {2 c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{63 a^4}+\frac {1}{9} a^4 c^2 x^8 \arctan (a x)^2 \sqrt {a^2 c x^2+c}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^2 \sqrt {a^2 c x^2+c}}{4032 a^4}+\frac {\left (a^2 c x^2+c\right )^{7/2}}{252 a^4 c}-\frac {23 \left (a^2 c x^2+c\right )^{5/2}}{7560 a^4}-\frac {115 c \left (a^2 c x^2+c\right )^{3/2}}{18144 a^4}+\frac {47 c^2 x \arctan (a x) \sqrt {a^2 c x^2+c}}{1344 a^3}-\frac {1}{36} a^3 c^2 x^7 \arctan (a x) \sqrt {a^2 c x^2+c} \]

[In]

Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

(-115*c^2*Sqrt[c + a^2*c*x^2])/(4032*a^4) - (115*c*(c + a^2*c*x^2)^(3/2))/(18144*a^4) - (23*(c + a^2*c*x^2)^(5
/2))/(7560*a^4) + (c + a^2*c*x^2)^(7/2)/(252*a^4*c) + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^3) -
(205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6048*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/1512
- (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/36 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^4) + (c^2
*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/21 + (19*a^2*
c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/9 - (((115*I)/
2016)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (
((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x
^2]) - (((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*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 267

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

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 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 5070

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[c^2*(d/f^2), Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 5072

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + (-Dist[b*f*(p/(c*m)), Int[(f*x)^(m - 1
)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[f^2*((m - 1)/(c^2*m)), Int[(f*x)^(m - 2)*((a +
b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && Gt
Q[m, 1]

Rubi steps \begin{align*} \text {integral}& = c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \\ & = c^2 \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = c^3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^9 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2}+\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (2 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (2 c^3\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}-\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (2 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (2 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{9} \left (8 a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{9} \left (2 a^5 c^3\right ) \int \frac {x^8 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3}-\frac {c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{10} c^3 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^3 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\left (4 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+\frac {\left (8 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^3\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {1}{21} \left (5 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{10} c^3 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^3\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {1}{21} \left (5 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^3\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{21} \left (16 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{36} \left (7 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{63} \left (16 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{36} \left (a^4 c^3\right ) \int \frac {x^7}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1320\) vs. \(2(578)=1156\).

Time = 6.72 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.28 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {\left (c+a^2 c x^2\right )^{5/2} \left (-48384 \left (50-32 \arctan (a x)^2+72 \cos (2 \arctan (a x))+160 \arctan (a x)^2 \cos (2 \arctan (a x))+22 \cos (4 \arctan (a x))-\frac {110 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {110 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {176 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \arctan (a x) \sin (2 \arctan (a x))-22 \arctan (a x) \sin (4 \arctan (a x))\right )+576 \left (1+a^2 x^2\right ) \left (4116+10944 \arctan (a x)^2+6262 \cos (2 \arctan (a x))-5376 \arctan (a x)^2 \cos (2 \arctan (a x))+2764 \cos (4 \arctan (a x))+6720 \arctan (a x)^2 \cos (4 \arctan (a x))+618 \cos (6 \arctan (a x))-\frac {10815 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-6489 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-2163 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-309 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {10815 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+6489 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+2163 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+309 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {19776 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{7/2}}+\frac {19776 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{7/2}}-1266 \arctan (a x) \sin (2 \arctan (a x))+360 \arctan (a x) \sin (4 \arctan (a x))-618 \arctan (a x) \sin (6 \arctan (a x))\right )-\left (1+a^2 x^2\right )^2 \left (657578-820224 \arctan (a x)^2+1083168 \cos (2 \arctan (a x))+3276288 \arctan (a x)^2 \cos (2 \arctan (a x))+576936 \cos (4 \arctan (a x))-580608 \arctan (a x)^2 \cos (4 \arctan (a x))+184160 \cos (6 \arctan (a x))+483840 \arctan (a x)^2 \cos (6 \arctan (a x))+32814 \cos (8 \arctan (a x))-\frac {2067282 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-1378188 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-590652 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-147663 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-16407 \arctan (a x) \cos (9 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {2067282 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+1378188 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+590652 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+147663 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+16407 \arctan (a x) \cos (9 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4200192 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{9/2}}+\frac {4200192 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{9/2}}+78444 \arctan (a x) \sin (2 \arctan (a x))-160452 \arctan (a x) \sin (4 \arctan (a x))+38172 \arctan (a x) \sin (6 \arctan (a x))-32814 \arctan (a x) \sin (8 \arctan (a x))\right )\right )}{46448640 a^4} \]

[In]

Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

((c + a^2*c*x^2)^(5/2)*(-48384*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan
[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[
a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan
[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]
*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^
(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]) + 576*(1 + a^2*x^2)*(4116 + 1094
4*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] +
6720*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x]
)])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Co
s[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])
] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*L
og[1 + I*E^(I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 309*ArcTan[a*
x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((19776*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x
^2)^(7/2) + ((19776*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) - 1266*ArcTan[a*x]*Sin[2*ArcTan[a*
x]] + 360*ArcTan[a*x]*Sin[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Sin[6*ArcTan[a*x]]) - (1 + a^2*x^2)^2*(657578 - 820
224*ArcTan[a*x]^2 + 1083168*Cos[2*ArcTan[a*x]] + 3276288*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 576936*Cos[4*ArcTa
n[a*x]] - 580608*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 184160*Cos[6*ArcTan[a*x]] + 483840*ArcTan[a*x]^2*Cos[6*Arc
Tan[a*x]] + 32814*Cos[8*ArcTan[a*x]] - (2067282*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] -
1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 590652*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Lo
g[1 - I*E^(I*ArcTan[a*x])] - 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 16407*ArcTan
[a*x]*Cos[9*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (2067282*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqr
t[1 + a^2*x^2] + 1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 590652*ArcTan[a*x]*Cos[
5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x]
)] + 16407*ArcTan[a*x]*Cos[9*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4200192*I)*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/(1 + a^2*x^2)^(9/2) + ((4200192*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(9/2) + 78444*Ar
cTan[a*x]*Sin[2*ArcTan[a*x]] - 160452*ArcTan[a*x]*Sin[4*ArcTan[a*x]] + 38172*ArcTan[a*x]*Sin[6*ArcTan[a*x]] -
32814*ArcTan[a*x]*Sin[8*ArcTan[a*x]])))/(46448640*a^4)

Maple [A] (verified)

Time = 10.11 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (20160 \arctan \left (a x \right )^{2} a^{8} x^{8}-5040 \arctan \left (a x \right ) a^{7} x^{7}+54720 a^{6} x^{6} \arctan \left (a x \right )^{2}+720 a^{6} x^{6}-12360 \arctan \left (a x \right ) a^{5} x^{5}+43200 a^{4} \arctan \left (a x \right )^{2} x^{4}+1608 a^{4} x^{4}-6150 \arctan \left (a x \right ) x^{3} a^{3}+2880 x^{2} \arctan \left (a x \right )^{2} a^{2}-94 a^{2} x^{2}+6345 x \arctan \left (a x \right ) a -5760 \arctan \left (a x \right )^{2}-6157\right )}{181440 a^{4}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{4032 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(309\)

[In]

int(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/181440*c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(20160*arctan(a*x)^2*a^8*x^8-5040*arctan(a*x)*a^7*x^7+54720*a^6*x^6
*arctan(a*x)^2+720*a^6*x^6-12360*arctan(a*x)*a^5*x^5+43200*a^4*arctan(a*x)^2*x^4+1608*a^4*x^4-6150*arctan(a*x)
*x^3*a^3+2880*x^2*arctan(a*x)^2*a^2-94*a^2*x^2+6345*x*arctan(a*x)*a-5760*arctan(a*x)^2-6157)-115/4032*c^2*(c*(
a*x-I)*(I+a*x))^(1/2)*(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)))/a^4/(a^2*x^2+1)^(1
/2)

Fricas [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2} \,d x } \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^2, x)

Sympy [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*(a**2*c*x**2+c)**(5/2)*atan(a*x)**2,x)

[Out]

Integral(x**3*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)

Maxima [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2} \,d x } \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*x^3*arctan(a*x)^2, x)

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^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. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

int(x^3*atan(a*x)^2*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x^3*atan(a*x)^2*(c + a^2*c*x^2)^(5/2), x)