3.4.40 \(\int \frac {x^2 \arctan (a x)^2}{(c+a^2 c x^2)^{3/2}} \, dx\) [340]

3.4.40.1 Optimal result

Integrand size = 24, antiderivative size = 349 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \arctan (a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}} \]

output

2*x/a^2/c/(a^2*c*x^2+c)^(1/2)-2*arctan(a*x)/a^3/c/(a^2*c*x^2+c)^(1/2)-x*ar 
ctan(a*x)^2/a^2/c/(a^2*c*x^2+c)^(1/2)-2*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/ 
2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)+2*I*arctan(a 
*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/c/(a^2 
*c*x^2+c)^(1/2)-2*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*( 
a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)-2*polylog(3,-I*(1+I*a*x)/(a^2*x 
^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)+2*polylog(3,I*(1+ 
I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)
 

3.4.40.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.65 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+a^2 x^2} \left (-\frac {2 a x}{\sqrt {1+a^2 x^2}}+\frac {2 \arctan (a x)}{\sqrt {1+a^2 x^2}}+\frac {a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}-\arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+\arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-2 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{a^3 c \sqrt {c \left (1+a^2 x^2\right )}} \]

input

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

output

-((Sqrt[1 + a^2*x^2]*((-2*a*x)/Sqrt[1 + a^2*x^2] + (2*ArcTan[a*x])/Sqrt[1 
+ a^2*x^2] + (a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - ArcTan[a*x]^2*Log[1 - 
 I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - (2*I) 
*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLo 
g[2, I*E^(I*ArcTan[a*x])] + 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 2*PolyL 
og[3, I*E^(I*ArcTan[a*x])]))/(a^3*c*Sqrt[c*(1 + a^2*x^2)]))
 

3.4.40.3 Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5499, 5425, 5423, 3042, 4669, 3011, 2720, 5433, 208, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}\)

\(\Big \downarrow \) 5423

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5433

\(\displaystyle -\frac {-2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 208

\(\displaystyle -\frac {\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^3 c \sqrt {a^2 c x^2+c}}\)

input

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

output

-(((-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^ 
2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]))/a^2) + (Sqrt[1 + a^2*x^2] 
*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLo 
g[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I* 
ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x 
])])))/(a^3*c*Sqrt[c + a^2*c*x^2])
 

3.4.40.3.1 Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

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

rule 5423

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ 
c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt 
Q[d, 0]
 

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[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 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

rule 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan 
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ 
[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

3.4.40.4 Maple [F]

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

input

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

output

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

3.4.40.5 Fricas [F]

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

input

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

output

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

3.4.40.6 Sympy [F]

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

input

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

output

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

3.4.40.7 Maxima [F]

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

input

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

output

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

3.4.40.8 Giac [F]

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

input

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

output

sage0*x
 

3.4.40.9 Mupad [F(-1)]

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

input

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

output

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