\(\int x (c+a^2 c x^2)^{5/2} \arctan (a x)^2 \, dx\) [325]

Optimal result

Integrand size = 22, antiderivative size = 387 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {5 c^2 \sqrt {c+a^2 c x^2}}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{28 a^2 \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}} \] Output:

5/56*c^2*(a^2*c*x^2+c)^(1/2)/a^2+5/252*c*(a^2*c*x^2+c)^(3/2)/a^2+1/105*(a^ 
2*c*x^2+c)^(5/2)/a^2-5/56*c^2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a-5/84*c*x 
*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a-1/21*x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)/ 
a+1/7*(a^2*c*x^2+c)^(7/2)*arctan(a*x)^2/a^2/c+5/28*I*c^3*(a^2*x^2+1)^(1/2) 
*arctan(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)-5/56*I*c^3*(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)+5/56*I*c^3*(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 [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1039\) vs. \(2(387)=774\).

Time = 6.77 (sec) , antiderivative size = 1039, normalized size of antiderivative = 2.68 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx =\text {Too large to display} \] Input:

Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]
 

Output:

(c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*(13440*(2 + 4*ArcTan[a*x]^2 + 2*Cos 
[2*ArcTan[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* 
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])] - ((4*I)*PolyLog[2, (-I)*E 
^(I*ArcTan[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]]) - 336*(1 + 
a^2*x^2)*(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]] - (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])] + (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])] - ((1 
76*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*P 
olyLog[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]]) + (1 + a^2*x^2)^2*(4116 
+ 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2 
*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] + 6720*ArcTan[a*x]^2*Cos[4*ArcTan[ 
a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcT...
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5465, 5413, 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.

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

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \int \left (a^2 c x^2+c\right )^{5/2} \arctan (a x)dx}{7 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \left (\frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)dx+\frac {1}{6} x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a}\right )}{7 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \left (\frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)dx+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a}\right )}{7 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a}\right )}{7 a}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a}\right )}{7 a}\)

\(\Big \downarrow \) 5421

\(\displaystyle \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {2 \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a}\right )}{7 a}\)

Input:

Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]
 

Output:

((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^2)/(7*a^2*c) - (2*(-1/30*(c + a^2*c*x^2 
)^(5/2)/a + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/6 + (5*c*(-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*Sqr 
t[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*PolyLog[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))/6)) 
/(7*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 = 4.86 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (360 a^{6} x^{6} \arctan \left (a x \right )^{2}-120 \arctan \left (a x \right ) a^{5} x^{5}+1080 a^{4} \arctan \left (a x \right )^{2} x^{4}+24 a^{4} x^{4}-390 \arctan \left (a x \right ) x^{3} a^{3}+1080 \arctan \left (a x \right )^{2} x^{2} a^{2}+98 a^{2} x^{2}-495 \arctan \left (a x \right ) a x +360 \arctan \left (a x \right )^{2}+299\right )}{2520 a^{2}}+\frac {5 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{56 a^{2} \sqrt {a^{2} x^{2}+1}}\)	\(275\)

Input:

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2520*c^2/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(360*a^6*x^6*arctan(a*x)^2-120*ar 
ctan(a*x)*a^5*x^5+1080*a^4*arctan(a*x)^2*x^4+24*a^4*x^4-390*arctan(a*x)*x^ 
3*a^3+1080*arctan(a*x)^2*x^2*a^2+98*a^2*x^2-495*arctan(a*x)*a*x+360*arctan 
(a*x)^2+299)+5/56*c^2*(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*d 
ilog(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]

\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{2} \,d x } \] Input:

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="fricas")
 

Output:

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*sqrt(a^2*c*x^2 + c)*arctan( 
a*x)^2, x)
 

Sympy [F]

\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \] Input:

integrate(x*(a**2*c*x**2+c)**(5/2)*atan(a*x)**2,x)
 

Output:

Integral(x*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)
 

Maxima [F]

\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{2} \,d x } \] Input:

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="maxima")
 

Output:

integrate((a^2*c*x^2 + c)^(5/2)*x*arctan(a*x)^2, x)
 

Giac [F(-2)]

Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(a^2*c*x^2+c)^(5/2)*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. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \] Input:

int(x*atan(a*x)^2*(c + a^2*c*x^2)^(5/2),x)
 

Output:

int(x*atan(a*x)^2*(c + a^2*c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\sqrt {c}\, c^{2} \left (\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{5}d x \right ) a^{4}+2 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{3}d x \right ) a^{2}+\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x d x \right ) \] Input:

int(x*(a^2*c*x^2+c)^(5/2)*atan(a*x)^2,x)
 

Output:

sqrt(c)*c**2*(int(sqrt(a**2*x**2 + 1)*atan(a*x)**2*x**5,x)*a**4 + 2*int(sq 
rt(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))