\(\int x (a+b \arctan (\frac {c}{x}))^2 \, dx\) [142]

Optimal result

Integrand size = 14, antiderivative size = 82 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b^2 c^2 \log \left (1+\frac {c^2}{x^2}\right )+b^2 c^2 \log (x) \] Output:

b*c*x*(a+b*arccot(x/c))+1/2*c^2*(a+b*arccot(x/c))^2+1/2*x^2*(a+b*arccot(x/ 
c))^2+1/2*b^2*c^2*ln(1+c^2/x^2)+b^2*c^2*ln(x)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{2} \left (a x (2 b c+a x)+2 b \left (b c x+a \left (c^2+x^2\right )\right ) \arctan \left (\frac {c}{x}\right )+b^2 \left (c^2+x^2\right ) \arctan \left (\frac {c}{x}\right )^2+b^2 c^2 \log \left (c^2+x^2\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5363, 5361, 5453, 5361, 243, 47, 14, 16, 5419}

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.

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

\(\Big \downarrow \) 5363

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

\(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5361

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

rule 5363

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], 
 x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplif 
y[(m + 1)/n]]
 

rule 5419

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

rule 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.28

method	result	size
parts	\(\frac {a^{2} x^{2}}{2}-b^{2} c^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+\arctan \left (\frac {c}{x}\right ) a b \,x^{2}-a b \,c^{2} \arctan \left (\frac {x}{c}\right )+a b c x\)	\(105\)
parallelrisch	\(\frac {\arctan \left (\frac {c}{x}\right )^{2} b^{2} x^{2}}{2}+\frac {\arctan \left (\frac {c}{x}\right )^{2} b^{2} c^{2}}{2}+\frac {b^{2} c^{2} \ln \left (c^{2}+x^{2}\right )}{2}+\arctan \left (\frac {c}{x}\right ) a b \,x^{2}+x \arctan \left (\frac {c}{x}\right ) b^{2} c +\arctan \left (\frac {c}{x}\right ) a b \,c^{2}+\frac {a^{2} x^{2}}{2}+a b c x -\frac {a^{2} c^{2}}{2}\)	\(107\)
derivativedivides	\(-c^{2} \left (-\frac {a^{2} x^{2}}{2 c^{2}}+b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )-\frac {a b \,x^{2} \arctan \left (\frac {c}{x}\right )}{c^{2}}-a b \arctan \left (\frac {c}{x}\right )-\frac {a b x}{c}\right )\)	\(113\)
default	\(-c^{2} \left (-\frac {a^{2} x^{2}}{2 c^{2}}+b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )-\frac {a b \,x^{2} \arctan \left (\frac {c}{x}\right )}{c^{2}}-a b \arctan \left (\frac {c}{x}\right )-\frac {a b x}{c}\right )\)	\(113\)
risch	\(\text {Expression too large to display}\)	\(13965\)

Input:

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

Output:

1/2*a^2*x^2-b^2*c^2*(-1/2/c^2*x^2*arctan(c/x)^2-1/2*arctan(c/x)^2-1/c*x*ar 
ctan(c/x)-1/2*ln(1+c^2/x^2)+ln(c/x))+arctan(c/x)*a*b*x^2-a*b*c^2*arctan(x/ 
c)+a*b*c*x
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=-a b c^{2} \arctan \left (\frac {x}{c}\right ) + \frac {1}{2} \, b^{2} c^{2} \log \left (c^{2} + x^{2}\right ) + a b c x + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{2} \, {\left (b^{2} c^{2} + b^{2} x^{2}\right )} \arctan \left (\frac {c}{x}\right )^{2} + {\left (b^{2} c x + a b x^{2}\right )} \arctan \left (\frac {c}{x}\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^{2} x^{2}}{2} + a b c^{2} \operatorname {atan}{\left (\frac {c}{x} \right )} + a b c x + a b x^{2} \operatorname {atan}{\left (\frac {c}{x} \right )} + \frac {b^{2} c^{2} \log {\left (c^{2} + x^{2} \right )}}{2} + \frac {b^{2} c^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2} + b^{2} c x \operatorname {atan}{\left (\frac {c}{x} \right )} + \frac {b^{2} x^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \arctan \left (\frac {c}{x}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \arctan \left (\frac {c}{x}\right ) - {\left (c \arctan \left (\frac {x}{c}\right ) - x\right )} c\right )} a b - \frac {1}{2} \, {\left ({\left (\arctan \left (\frac {x}{c}\right )^{2} - \log \left (c^{2} + x^{2}\right )\right )} c^{2} + 2 \, {\left (c \arctan \left (\frac {x}{c}\right ) - x\right )} c \arctan \left (\frac {c}{x}\right )\right )} b^{2} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^2\,x^2}{2}+\frac {b^2\,c^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,c^2\,\ln \left (c^2+x^2\right )}{2}+\frac {b^2\,x^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+a\,b\,c^2\,\mathrm {atan}\left (\frac {c}{x}\right )+a\,b\,x^2\,\mathrm {atan}\left (\frac {c}{x}\right )+b^2\,c\,x\,\mathrm {atan}\left (\frac {c}{x}\right )+a\,b\,c\,x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {\mathit {atan} \left (\frac {c}{x}\right )^{2} b^{2} c^{2}}{2}+\frac {\mathit {atan} \left (\frac {c}{x}\right )^{2} b^{2} x^{2}}{2}+\mathit {atan} \left (\frac {c}{x}\right ) a b \,c^{2}+\mathit {atan} \left (\frac {c}{x}\right ) a b \,x^{2}+\mathit {atan} \left (\frac {c}{x}\right ) b^{2} c x +\frac {\mathrm {log}\left (c^{2}+x^{2}\right ) b^{2} c^{2}}{2}+\frac {a^{2} x^{2}}{2}+a b c x \] Input:

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

Output:

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