\(\int x (c+a^2 c x^2)^{3/2} \arctan (a x)^2 \, dx\) [317]

Optimal result

Integrand size = 22, antiderivative size = 334 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {3 c \sqrt {c+a^2 c x^2}}{20 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)}{20 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{10 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{5 a^2 c}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{10 a^2 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {c+a^2 c x^2}} \] Output:

3/20*c*(a^2*c*x^2+c)^(1/2)/a^2+1/30*(a^2*c*x^2+c)^(3/2)/a^2-3/20*c*x*(a^2* 
c*x^2+c)^(1/2)*arctan(a*x)/a-1/10*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a+1/5* 
(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2/a^2/c+3/10*I*c^2*(a^2*x^2+1)^(1/2)*arcta 
n(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^2/(a^2*c*x^2+c)^(1/2)-3/2 
0*I*c^2*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^ 
2/(a^2*c*x^2+c)^(1/2)+3/20*I*c^2*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^( 
1/2)/(1-I*a*x)^(1/2))/a^2/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.85 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.80 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \left (80 \left (2+4 \arctan (a x)^2+2 \cos (2 \arctan (a x))-\frac {3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )-\left (1+a^2 x^2\right ) \left (50-32 \arctan (a x)^2+72 \cos (2 \arctan (a x))+160 \arctan (a x)^2 \cos (2 \arctan (a x))+22 \cos (4 \arctan (a x))-\frac {110 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {110 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {176 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \arctan (a x) \sin (2 \arctan (a x))-22 \arctan (a x) \sin (4 \arctan (a x))\right )\right )}{960 a^2} \] Input:

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

Output:

(c*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*(80*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*Ar 
cTan[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*ArcTa 
n[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])] - ((4*I)*PolyLog[2, (-I)*E^(I*A 
rcTan[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]]) - (1 + a^2*x^2)* 
(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*A 
rcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcT 
an[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*Arc 
Tan[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)*Pol 
yLog[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*a^2)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5465, 5413, 5413, 5425, 5421}

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.

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

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {2 \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)dx}{5 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {2 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)dx+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )}{5 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {2 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )}{5 a}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {2 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )}{5 a}\)

\(\Big \downarrow \) 5421

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {2 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )}{5 a}\)

Input:

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

Output:

((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(5*a^2*c) - (2*(-1/12*(c + a^2*c*x^2 
)^(3/2)/a + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 + (3*c*(-1/2*Sqrt[c + 
a^2*c*x^2]/a + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 + (c*Sqrt[1 + a^2*x^2 
]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*Pol 
yLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt 
[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(2*Sqrt[c + a^2*c*x^2])))/4))/(5*a)
 

Defintions of rubi rules used

rule 5413

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

rule 5421

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]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
GtQ[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]
 

Maple [A] (verified)

Time = 2.05 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+24 \arctan \left (a x \right )^{2} x^{2} a^{2}+2 a^{2} x^{2}-15 \arctan \left (a x \right ) a x +12 \arctan \left (a x \right )^{2}+11\right )}{60 a^{2}}+\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{20 a^{2} \sqrt {a^{2} x^{2}+1}}\)	\(237\)

Input:

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

Output:

1/60*c/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(12*a^4*arctan(a*x)^2*x^4-6*arctan(a* 
x)*x^3*a^3+24*arctan(a*x)^2*x^2*a^2+2*a^2*x^2-15*arctan(a*x)*a*x+12*arctan 
(a*x)^2+11)+3/20*c*(c*(a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)*ln(1+I*(1+I*a*x) 
/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*dilo 
g(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2) 
))/a^2/(a^2*x^2+1)^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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