\(\int (a+b \arctan (\frac {c}{x}))^2 \, dx\) [143]

Optimal result

Integrand size = 12, antiderivative size = 83 \[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=i c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-2 b c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2 c}{c+i x}\right )+i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right ) \] Output:

I*c*(a+b*arccot(x/c))^2+x*(a+b*arccot(x/c))^2-2*b*c*(a+b*arccot(x/c))*ln(2 
*c/(c+I*x))+I*b^2*c*polylog(2,1-2*c/(c+I*x))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.27 \[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=b^2 (i c+x) \arctan \left (\frac {c}{x}\right )^2+2 b \arctan \left (\frac {c}{x}\right ) \left (a x-b c \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+a \left (a x+b c \log \left (1+\frac {c^2}{x^2}\right )-2 b c \log \left (\frac {c}{x}\right )\right )+i b^2 c \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right ) \] Input:

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

Output:

b^2*(I*c + x)*ArcTan[c/x]^2 + 2*b*ArcTan[c/x]*(a*x - b*c*Log[1 - E^((2*I)* 
ArcTan[c/x])]) + a*(a*x + b*c*Log[1 + c^2/x^2] - 2*b*c*Log[c/x]) + I*b^2*c 
*PolyLog[2, E^((2*I)*ArcTan[c/x])]
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5349, 5346, 27, 5456, 27, 5380, 27, 2849, 2752}

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.

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

\(\Big \downarrow \) 5349

\(\displaystyle \int \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2dx\)

\(\Big \downarrow \) 5346

\(\displaystyle \frac {2 b \int \frac {c^2 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{c^2+x^2}dx}{c}+x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\)

\(\Big \downarrow \) 27

\(\displaystyle 2 b c \int \frac {x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{c^2+x^2}dx+x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\)

\(\Big \downarrow \) 5456

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\frac {\int \frac {c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{i c-x}dx}{c}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\int \frac {a+b \cot ^{-1}\left (\frac {x}{c}\right )}{i c-x}dx\right )\)

\(\Big \downarrow \) 5380

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (-\frac {b \int \frac {c^2 \log \left (\frac {2 c}{c+i x}\right )}{c^2+x^2}dx}{c}+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (-b c \int \frac {\log \left (\frac {2 c}{c+i x}\right )}{c^2+x^2}dx+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (i b c \int \frac {\log \left (\frac {2 c}{c+i x}\right )}{1-\frac {2 c}{c+i x}}d\frac {1}{c+i x}+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+2 b c \left (\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right )\right )\)

Input:

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

Output:

x*(a + b*ArcCot[x/c])^2 + 2*b*c*(((I/2)*(a + b*ArcCot[x/c])^2)/b - (a + b* 
ArcCot[x/c])*Log[(2*c)/(c + I*x)] + (I/2)*b*PolyLog[2, 1 - (2*c)/(c + I*x) 
])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5346

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p   Int[x^n*((a + b*ArcCot[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5349

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[(a + b*A 
rcCot[1/(x^n*c)])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
 

rule 5380

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

rule 5456

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (79 ) = 158\).

Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.71

method	result	size
parts	\(a^{2} x -b^{2} c \left (-\frac {x \arctan \left (\frac {c}{x}\right )^{2}}{c}-\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{2}+i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )-i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )+i \operatorname {dilog}\left (1+\frac {i c}{x}\right )-i \operatorname {dilog}\left (1-\frac {i c}{x}\right )\right )-2 a b c \left (-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\)	\(308\)
derivativedivides	\(-c \left (-\frac {a^{2} x}{c}+b^{2} \left (-\frac {x \arctan \left (\frac {c}{x}\right )^{2}}{c}-\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{2}+i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )-i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )+i \operatorname {dilog}\left (1+\frac {i c}{x}\right )-i \operatorname {dilog}\left (1-\frac {i c}{x}\right )\right )+2 a b \left (-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\right )\)	\(312\)
default	\(-c \left (-\frac {a^{2} x}{c}+b^{2} \left (-\frac {x \arctan \left (\frac {c}{x}\right )^{2}}{c}-\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{2}+i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )-i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )+i \operatorname {dilog}\left (1+\frac {i c}{x}\right )-i \operatorname {dilog}\left (1-\frac {i c}{x}\right )\right )+2 a b \left (-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\right )\)	\(312\)
risch	\(\text {Expression too large to display}\)	\(23471\)

Input:

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

Output:

a^2*x-b^2*c*(-1/c*x*arctan(c/x)^2-arctan(c/x)*ln(1+c^2/x^2)+2*ln(c/x)*arct 
an(c/x)-1/2*I*(ln(c/x-I)*ln(1+c^2/x^2)-1/2*ln(c/x-I)^2-dilog(-1/2*I*(c/x+I 
))-ln(c/x-I)*ln(-1/2*I*(c/x+I)))+1/2*I*(ln(c/x+I)*ln(1+c^2/x^2)-1/2*ln(c/x 
+I)^2-dilog(1/2*I*(c/x-I))-ln(c/x+I)*ln(1/2*I*(c/x-I)))+I*ln(c/x)*ln(1+I*c 
/x)-I*ln(c/x)*ln(1-I*c/x)+I*dilog(1+I*c/x)-I*dilog(1-I*c/x))-2*a*b*c*(-1/c 
*x*arctan(c/x)-1/2*ln(1+c^2/x^2)+ln(c/x))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(2*x*arctan(c/x) + c*log(c^2 + x^2))*a*b + 1/16*(12*c*arctan(c/x)^2*arctan 
(x/c) + 4*(3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/c)*c^2 + 4*x*arct 
an2(c, x)^2 + 16*c^2*integrate(1/16*log(c^2 + x^2)^2/(c^2 + x^2), x) - x*l 
og(c^2 + x^2)^2 + 128*c*integrate(1/16*x*arctan(c/x)/(c^2 + x^2), x) + 192 
*integrate(1/16*x^2*arctan(c/x)^2/(c^2 + x^2), x) + 16*integrate(1/16*x^2* 
log(c^2 + x^2)^2/(c^2 + x^2), x) + 64*integrate(1/16*x^2*log(c^2 + x^2)/(c 
^2 + x^2), x))*b^2 + a^2*x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

2*atan(c/x)*a*b*x + int(atan(c/x)**2,x)*b**2 + log(c**2 + x**2)*a*b*c + a* 
*2*x