\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)^2}{x} \, dx\) [327]

Optimal result

Integrand size = 24, antiderivative size = 605 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x} \, dx=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

1/30*c*(a^2*c*x^2+c)^(3/2)-1/10*a*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)+1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+
1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2+149/30*I*c^3*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^
2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-2*c^3*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2
*c*x^2+c)^(1/2)+2*I*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1
/2)-2*I*c^3*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-149/60*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^2*c*x^2+c)^(1/2)+149/60*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^2*c*x^2+c)^(1/2)-2*c^3*polylog(3,-(1+I*a*x)/(a^2*x^2+
1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+2*c^3*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)
/(a^2*c*x^2+c)^(1/2)+29/60*c^2*(a^2*c*x^2+c)^(1/2)-29/60*a*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+c^2*arctan(a*
x)^2*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5070, 5078, 5076, 4268, 2611, 2320, 6724, 5050, 5010, 5006, 4998} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x} \, dx=-\frac {2 c^3 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {2 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {a^2 c x^2+c}}+c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {29}{60} a c^2 x \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{3} c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {1}{10} a c x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}+\frac {1}{5} \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {149 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {29}{60} c^2 \sqrt {a^2 c x^2+c}+\frac {1}{30} c \left (a^2 c x^2+c\right )^{3/2} \]

[In]

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

[Out]

(29*c^2*Sqrt[c + a^2*c*x^2])/60 + (c*(c + a^2*c*x^2)^(3/2))/30 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
60 - (a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/10 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (c*(c + a^2*c*x^2)
^(3/2)*ArcTan[a*x]^2)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/5 + (((149*I)/30)*c^3*Sqrt[1 + a^2*x^2]*ArcTan
[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Ar
cTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcT
an[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[
c + a^2*c*x^2] - (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[
c + a^2*c*x^2] + (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c +
a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c^3*Sqrt[1 + a
^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4998

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

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 5076

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

Rule 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \\ & = \frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {1}{5} (2 a c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx+c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = \frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {1}{10} \left (3 a c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx-\frac {1}{3} \left (2 a c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+c^3 \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {1}{20} \left (3 a c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\left (2 a c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {149 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 )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {149 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 )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {149 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 )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {149 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 )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 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 )}{60 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.17 (sec) , antiderivative size = 889, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x} \, dx=c^2 \sqrt {c \left (1+a^2 x^2\right )} \left (\arctan (a x)^2+\frac {\arctan (a x)^2 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\frac {2 \left (\arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}\right )+\frac {1}{6} c^2 \left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2+4 \arctan (a x)^2+2 \cos (2 \arctan (a x))-\frac {3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )-\frac {1}{960} c^2 \left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \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 ) \]

[In]

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

[Out]

c^2*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]^2 + (ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*
x])]))/Sqrt[1 + a^2*x^2] - (2*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(
PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/Sqrt[1 + a^2*x^2] + ((2*I)*ArcTan[a*x]
*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (2*(-PolyLog[3, -E^(I*A
rcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (c^2*(1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(
2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] -
ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqr
t[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*
Sin[2*ArcTan[a*x]]))/6 - (c^2*(1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(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]]))/960

Maple [A] (verified)

Time = 3.02 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+44 x^{2} \arctan \left (a x \right )^{2} a^{2}+2 a^{2} x^{2}-35 x \arctan \left (a x \right ) a +92 \arctan \left (a x \right )^{2}+31\right )}{60}+\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (60 i \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-60 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-149 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+149 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+120 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-149 \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+149 \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 \sqrt {a^{2} x^{2}+1}}\)	\(404\)

[In]

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

[Out]

1/60*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(12*a^4*arctan(a*x)^2*x^4-6*arctan(a*x)*x^3*a^3+44*x^2*arctan(a*x)^2*a^2+2*
a^2*x^2-35*x*arctan(a*x)*a+92*arctan(a*x)^2+31)+1/60*I*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(60*I*arctan(a*x)^2*ln((1
+I*a*x)/(a^2*x^2+1)^(1/2)+1)-60*I*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*I*arctan(a*x)*ln(1+I*(1+
I*a*x)/(a^2*x^2+1)^(1/2))+149*I*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*arctan(a*x)*polylog(2,-(1+
I*a*x)/(a^2*x^2+1)^(1/2))-120*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*I*polylog(3,-(1+I*a*x)/(a
^2*x^2+1)^(1/2))-120*I*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+149*d
ilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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