\(\int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\) [310]

Optimal result

Integrand size = 21, antiderivative size = 340 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}} \] Output:

-(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a+1/2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2 
-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 
/(a^2*c*x^2+c)^(1/2)+c^(1/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/a+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/(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/(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/(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/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.59 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (2 \coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-2 \sqrt {1+a^2 x^2} \arctan (a x)+a x \sqrt {1+a^2 x^2} \arctan (a x)^2-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{2 a \sqrt {1+a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.67, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {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.

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

\(\Big \downarrow \) 5415

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((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]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (c*Sqrt[ 
1 + a^2*x^2]*((-2*I)*ArcTan[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])
 

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 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 = 1.97 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (\arctan \left (a x \right ) a x -2\right )}{2 a}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 a \sqrt {a^{2} x^{2}+1}}\)	\(268\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,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. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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