\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)^2}{x^3} \, dx\) [321]

Optimal result

Integrand size = 24, antiderivative size = 567 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx=-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

-a^2*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))+4*I*a^2*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^2*c*x^2+c)^(1/2)-3*a^2*c^2*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)+3*I*a^2*c^2*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)-3*I*a^2*c^2*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*a^2*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^2*c*x^2+c)^(1/2)+
2*I*a^2*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^2*c*x^2+c)^(1/2)-3*a^2*c^2*polyl
og(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)+3*a^2*c^2*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)-a*c*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x+a^2*c*arctan(a*x)^2*(a
^2*c*x^2+c)^(1/2)-1/2*c*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/x^2

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5070, 5082, 5064, 272, 65, 214, 5078, 5076, 4268, 2611, 2320, 6724, 5050, 5010, 5006} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx=-\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {4 i a^2 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {a c \arctan (a x) \sqrt {a^2 c x^2+c}}{x}-\frac {c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 x^2}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {2 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {2 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

-((a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (c*Sqrt[c + a^2*c*x^2]*
ArcTan[a*x]^2)/(2*x^2) + ((4*I)*a^2*c^2*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] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x
^2] - a^2*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[
2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*Arc
Tan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^2*c^2*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*I)*a^2*c^2*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] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3
*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 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 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 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -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 5082

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] + (-Dist[b*c*(p/(f*(m + 1))), Int[(f*x
)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[c^2*((m + 2)/(f^2*(m + 1))), 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] && G
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

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 {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x} \, dx \\ & = c^2 \int \frac {\arctan (a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\left (a c^2\right ) \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx-\left (2 a^3 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+2 \frac {\left (a^2 c^2 \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 {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\left (a^2 c^2\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+2 \frac {\left (a^2 c^2 \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 (2 a^3 c^2 \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 {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 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 )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {1}{2} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \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 {\left (2 a^2 c^2 \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 a^2 c^2 \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}}\right ) \\ & = -\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^2 \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 a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+c \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 {\left (2 i a^2 c^2 \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 a^2 c^2 \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}}\right )+\frac {\left (a^2 c^2 \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 (a^2 c^2 \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 {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^2 \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}}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )-\frac {i a^2 c^2 \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 {i a^2 c^2 \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 a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (i a^2 c^2 \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 (i a^2 c^2 \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}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 {\left (2 a^2 c^2 \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 a^2 c^2 \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}}\right ) \\ & = -\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^2 \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}}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )-\frac {i a^2 c^2 \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 {i a^2 c^2 \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 a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^2 \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 (a^2 c^2 \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 {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^2 \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}}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )-\frac {i a^2 c^2 \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 {i a^2 c^2 \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 a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 2.48 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx=\frac {a^2 c \sqrt {c+a^2 c x^2} \left (-4 \arctan (a x)-4 \arctan (a x) \cot ^2\left (\frac {1}{2} \arctan (a x)\right )+4 a x \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+12 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-16 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-i e^{i \arctan (a x)}\right )+16 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+i e^{i \arctan (a x)}\right )-12 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+8 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+24 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-16 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+16 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-24 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-24 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+24 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+\arctan (a x)^2 \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{8 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(a^2*c*Sqrt[c + a^2*c*x^2]*(-4*ArcTan[a*x] - 4*ArcTan[a*x]*Cot[ArcTan[a*x]/2]^2 + 4*a*x*ArcTan[a*x]^2*Csc[ArcT
an[a*x]/2]^2 - ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log
[1 - E^(I*ArcTan[a*x])] - 16*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - I*E^(I*ArcTan[a*x])] + 16*ArcTan[a*x]*Cot[
ArcTan[a*x]/2]*Log[1 + I*E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] +
8*Cot[ArcTan[a*x]/2]*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[
a*x])] - (16*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (16*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, I
*E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*Cot[ArcTan[a*x]
/2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*Cot[ArcTan[a*x]/2]*PolyLog[3, E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Csc[A
rcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 a^{2} \arctan \left (a x \right ) x^{2}-2 a x -\arctan \left (a x \right )\right )}{2 x^{2}}-\frac {c \,a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\)	\(412\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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