\(\int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx\) [414]

Optimal result

Integrand size = 22, antiderivative size = 373 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}} \] Output:

(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2-1/2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^ 
2/a+I*c*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^ 
2/a^2/(a^2*c*x^2+c)^(1/2)+1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/a^2/c-c^(1 
/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/a^2-I*c*(a^2*x^2+1)^(1/2)*arc 
tan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^2/(a^2*c*x^2+c)^(1/2) 
+I*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2) 
)/a^2/(a^2*c*x^2+c)^(1/2)+c*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2* 
x^2+1)^(1/2))/a^2/(a^2*c*x^2+c)^(1/2)-c*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I 
*a*x)/(a^2*x^2+1)^(1/2))/a^2/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.55 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-\frac {12 \left (\coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+\left (1+a^2 x^2\right ) \arctan (a x) \left (6+4 \arctan (a x)^2+6 \cos (2 \arctan (a x))-3 \arctan (a x) \sin (2 \arctan (a x))\right )\right )}{12 a^2} \] Input:

Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
 

Output:

(Sqrt[c + a^2*c*x^2]*((-12*(ArcCoth[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[E^ 
(I*ArcTan[a*x])]*ArcTan[a*x]^2 + I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan 
[a*x])] - I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)* 
E^(I*ArcTan[a*x])] + PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + 
 (1 + a^2*x^2)*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3 
*ArcTan[a*x]*Sin[2*ArcTan[a*x]])))/(12*a^2)
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.71, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5465, 5415, 224, 219, 5425, 5423, 3042, 4669, 3011, 2720, 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.

\(\displaystyle \int x \arctan (a x)^3 \sqrt {a^2 c x^2+c} \, dx\)

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5415

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
 

Output:

((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*a^2*c) - (-((Sqrt[c + a^2*c*x^2]* 
ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + (Sqrt[c]*ArcTa 
nh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (c*Sqrt[1 + a^2*x^2]*((-2*I)*Ar 
cTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[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])])))/(2*a 
*Sqrt[c + a^2*c*x^2]))/a
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 5415

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

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 5465

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] - Simp[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 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]
 

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 \arctan \left (a x \right )^{2} x^{2} a^{2}-3 \arctan \left (a x \right ) a x +2 \arctan \left (a x \right )^{2}+6\right )}{6 a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2 i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2} x^{2}+1}}\)	\(370\)

Input:

int(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/6/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)*(2*arctan(a*x)^2*x^2*a^2-3*a 
rctan(a*x)*a*x+2*arctan(a*x)^2+6)+1/6*(c*(a*x-I)*(a*x+I))^(1/2)*(I*arctan( 
a*x)^3-3*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x) 
*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2 
+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+1/6*(c*(a*x-I)*(a*x+I))^(1/2)*(-I*arctan 
(a*x)^3+3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x 
)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(3,-I*(1+I*a*x)/(a^2* 
x^2+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+2*I/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*arc 
tan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)
 

Fricas [F]

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

integrate(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\sqrt {c}\, \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x d x \right ) \] Input:

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

Output:

sqrt(c)*int(sqrt(a**2*x**2 + 1)*atan(a*x)**3*x,x)