\(\int x^3 (c+a^2 c x^2)^{3/2} \arctan (a x)^2 \, dx\) [315]

Optimal result

Integrand size = 24, antiderivative size = 476 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=-\frac {17 c \sqrt {c+a^2 c x^2}}{280 a^4}-\frac {17 \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}+\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {17 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{140 a^4 \sqrt {c+a^2 c x^2}}+\frac {17 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {17 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {c+a^2 c x^2}} \]

[Out]

-17/1260*(a^2*c*x^2+c)^(3/2)/a^4+1/105*(a^2*c*x^2+c)^(5/2)/a^4/c-17/140*I*c^2*arctan(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)+17/280*I*c^2*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)-17/280*I*c^2*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)-17/280*c*(a^2*c*x^2+c)^(1/2)/a^4+3/56*c*x*arctan(a*x)*(a^2*c*x^2+c)
^(1/2)/a^3-23/420*c*x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a-1/21*a*c*x^5*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-2/35*c*
arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^4+1/35*c*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+8/35*c*x^4*arctan(a*x)^
2*(a^2*c*x^2+c)^(1/2)+1/7*a^2*c*x^6*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 2.90 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 75, 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} \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{35 a^2}+\frac {1}{7} a^2 c x^6 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {1}{21} a c x^5 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {8}{35} c x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {23 c x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{420 a}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{140 a^4 \sqrt {a^2 c x^2+c}}-\frac {2 c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{35 a^4}+\frac {17 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{105 a^4 c}-\frac {17 \left (a^2 c x^2+c\right )^{3/2}}{1260 a^4}-\frac {17 c \sqrt {a^2 c x^2+c}}{280 a^4}+\frac {3 c x \arctan (a x) \sqrt {a^2 c x^2+c}}{56 a^3} \]

[In]

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

[Out]

(-17*c*Sqrt[c + a^2*c*x^2])/(280*a^4) - (17*(c + a^2*c*x^2)^(3/2))/(1260*a^4) + (c + a^2*c*x^2)^(5/2)/(105*a^4
*c) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(420*a) -
(a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^4) + (c*x^2*Sqrt[
c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/35 + (a^2*c*x^6*Sqrt[c +
a^2*c*x^2]*ArcTan[a*x]^2)/7 - (((17*I)/140)*c^2*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]) + (((17*I)/280)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqr
t[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((17*I)/280)*c^2*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 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = c^2 \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^7 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2}+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (2 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+2 \left (\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{5} \left (4 c^2\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (2 a c^2\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (6 a^2 c^2\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^2\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{35} \left (24 c^2\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\left (4 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c^2 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+2 \left (-\frac {c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{10} c^2 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^2\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}\right )+\frac {1}{21} \left (5 a c^2\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^2\right ) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^2\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c \sqrt {c+a^2 c x^2}}{3 a^4}-\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{84} \left (5 c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (3 c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {5 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{20} c^2 \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {\left (4 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (16 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (3 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {\left (4 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}\right )-\frac {\left (16 c^2\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (5 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{28 a}-\frac {\left (9 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {\left (16 c^2\right ) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {1}{42} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {c \sqrt {c+a^2 c x^2}}{3 a^4}-\frac {131 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {118 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {10 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {1}{168} \left (5 c^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{70} \left (3 c^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (5 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^3}+\frac {\left (9 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}+\frac {\left (8 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (32 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (5 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^2}+\frac {\left (9 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^2}+\frac {\left (8 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {1}{42} \left (a^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+2 \left (-\frac {5 c \sqrt {c+a^2 c x^2}}{12 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{20} c^2 \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}\right ) \\ & = \frac {139 c \sqrt {c+a^2 c x^2}}{168 a^4}-\frac {2 \left (c+a^2 c x^2\right )^{3/2}}{63 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}-\frac {131 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {118 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {10 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {31 c \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {89 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}\right )-\frac {1}{168} \left (5 c^2\right ) \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {1}{70} \left (3 c^2\right ) \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{56 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{70 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (32 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {817 c \sqrt {c+a^2 c x^2}}{840 a^4}-\frac {101 \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}-\frac {131 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {118 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{420 a^4 \sqrt {c+a^2 c x^2}}+\frac {2543 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {31 c \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {89 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.92 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.67 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \left (-168 \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 )+\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 )\right )}{161280 a^4} \]

[In]

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

[Out]

(c*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2]*(-168*(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)*PolyLog[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]]) + (1 + a^2*x^2)*(4
116 + 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTa
n[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*A
rcTan[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*ArcT
an[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*Ar
cTan[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*ArcT
an[a*x]]*Log[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]])))/(161280*a^4)

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.57

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (360 a^{6} x^{6} \arctan \left (a x \right )^{2}-120 \arctan \left (a x \right ) a^{5} x^{5}+576 a^{4} \arctan \left (a x \right )^{2} x^{4}+24 a^{4} x^{4}-138 \arctan \left (a x \right ) x^{3} a^{3}+72 x^{2} \arctan \left (a x \right )^{2} a^{2}+14 a^{2} x^{2}+135 x \arctan \left (a x \right ) a -144 \arctan \left (a x \right )^{2}-163\right )}{2520 a^{4}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{280 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(271\)

[In]

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

[Out]

1/2520*c/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(360*a^6*x^6*arctan(a*x)^2-120*arctan(a*x)*a^5*x^5+576*a^4*arctan(a*x)^
2*x^4+24*a^4*x^4-138*arctan(a*x)*x^3*a^3+72*x^2*arctan(a*x)^2*a^2+14*a^2*x^2+135*x*arctan(a*x)*a-144*arctan(a*
x)^2-163)-17/280*c*(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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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