3.2.50 \(\int (a+b \arctan (\frac {c}{x}))^3 \, dx\) [150]

3.2.50.1 Optimal result

Integrand size = 12, antiderivative size = 119 \[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=i c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2 c}{c+i x}\right )+3 i b^2 c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right )-\frac {3}{2} b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c}{c+i x}\right ) \]

output

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

3.2.50.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.81 \[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=a^3 x+3 a^2 b x \arctan \left (\frac {c}{x}\right )+\frac {3}{2} a^2 b c \log \left (c^2+x^2\right )-3 a b^2 \left (-\left ((i c+x) \arctan \left (\frac {c}{x}\right )^2\right )+2 c \arctan \left (\frac {c}{x}\right ) \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )-i c \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )-\frac {1}{8} b^3 \left (-i c \pi ^3+8 i c \arctan \left (\frac {c}{x}\right )^3-8 x \arctan \left (\frac {c}{x}\right )^3+24 c \arctan \left (\frac {c}{x}\right )^2 \log \left (1-e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )+24 i c \arctan \left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )+12 c \operatorname {PolyLog}\left (3,e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \]

input

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

output

a^3*x + 3*a^2*b*x*ArcTan[c/x] + (3*a^2*b*c*Log[c^2 + x^2])/2 - 3*a*b^2*(-( 
(I*c + x)*ArcTan[c/x]^2) + 2*c*ArcTan[c/x]*Log[1 - E^((2*I)*ArcTan[c/x])] 
- I*c*PolyLog[2, E^((2*I)*ArcTan[c/x])]) - (b^3*((-I)*c*Pi^3 + (8*I)*c*Arc 
Tan[c/x]^3 - 8*x*ArcTan[c/x]^3 + 24*c*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcT 
an[c/x])] + (24*I)*c*ArcTan[c/x]*PolyLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c 
*PolyLog[3, E^((-2*I)*ArcTan[c/x])]))/8
 

3.2.50.3 Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5349, 5346, 27, 5456, 27, 5380, 27, 5530, 7164}

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.

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

\(\Big \downarrow \) 5349

\(\displaystyle \int \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3dx\)

\(\Big \downarrow \) 5346

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5456

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5380

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+3 b c \left (-\frac {2 b \int \frac {c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2 c}{c+i x}\right )}{c^2+x^2}dx}{c}+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{3 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+3 b c \left (-2 b c \int \frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2 c}{c+i x}\right )}{c^2+x^2}dx+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{3 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\right )\)

\(\Big \downarrow \) 5530

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+3 b c \left (-2 b c \left (-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right )}{c^2+x^2}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{2 c}\right )+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{3 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+3 b c \left (-2 b c \left (\frac {b \operatorname {PolyLog}\left (3,1-\frac {2 c}{c+i x}\right )}{4 c}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{2 c}\right )+\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{3 b}-\log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2\right )\)

input

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

output

x*(a + b*ArcCot[x/c])^3 + 3*b*c*(((I/3)*(a + b*ArcCot[x/c])^3)/b - (a + b* 
ArcCot[x/c])^2*Log[(2*c)/(c + I*x)] - 2*b*c*(((-1/2*I)*(a + b*ArcCot[x/c]) 
*PolyLog[2, 1 - (2*c)/(c + I*x)])/c + (b*PolyLog[3, 1 - (2*c)/(c + I*x)])/ 
(4*c)))
 

3.2.50.3.1 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 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]
 

rule 5530

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

3.2.50.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 102.88 (sec) , antiderivative size = 2028, normalized size of antiderivative = 17.04

method	result	size
parts	\(\text {Expression too large to display}\)	\(2028\)
derivativedivides	\(\text {Expression too large to display}\)	\(2031\)
default	\(\text {Expression too large to display}\)	\(2031\)

input

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

output

x*a^3-b^3*c*(-1/c*x*arctan(c/x)^3+3*ln(c/x)*arctan(c/x)^2-3/2*arctan(c/x)^ 
2*ln(1+c^2/x^2)+3*arctan(c/x)^2*ln((1+I*c/x)/(1+c^2/x^2)^(1/2))-3*arctan(c 
/x)^2*ln((1+I*c/x)^2/(1+c^2/x^2)-1)-I*arctan(c/x)^3+3/4*(-I*Pi*csgn(I/((1+ 
I*c/x)^2/(1+c^2/x^2)+1)^2)*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))*csgn(I*(1+I*c/x 
)^2/(1+c^2/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)+2*I*Pi*csgn(((1+I*c/x)^2/(1 
+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^3+2*I*Pi*csgn(I*((1+I*c/x)^2/(1+ 
c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^3+2*I*Pi*csgn(I*(1+I*c/x)/(1+c^2/ 
x^2)^(1/2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))^2-2*I*Pi*csgn(I*((1+I*c/x)^2/( 
1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(((1+I*c/x)^2/(1+c^2/x^2)-1 
)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2-2*I*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1) 
)*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2-I*Pi*c 
sgn(I*(1+I*c/x)/(1+c^2/x^2)^(1/2))^2*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))+I*Pi* 
csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1))^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2 
)+I*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^3-2*I*Pi*csgn(I*((1+I*c/x)^2/ 
(1+c^2/x^2)+1))*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^2-2*I*Pi*csgn(((1+I* 
c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2+2*I*Pi*csgn(I*((1+I*c 
/x)^2/(1+c^2/x^2)-1))*csgn(I/((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(I*((1+I*c/x 
)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))+2*I*Pi+I*Pi*csgn(I*(1+I*c/ 
x)^2/(1+c^2/x^2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+ 
1)^2)^2-I*Pi*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))^3+2*I*Pi*csgn(I*((1+I*c/x)...
 

3.2.50.5 Fricas [F]

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

input

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

output

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

3.2.50.6 Sympy [F]

\[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]

input

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

output

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

3.2.50.7 Maxima [F]

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

input

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

output

7/8*b^3*c*arctan(c/x)^3*arctan(x/c) + 3*a*b^2*c*arctan(c/x)^2*arctan(x/c) 
+ 1/8*b^3*x*arctan2(c, x)^3 - 3/32*b^3*x*arctan2(c, x)*log(c^2 + x^2)^2 + 
(3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/c)*a*b^2*c^2 + 7/32*(6*arct 
an(c/x)^2*arctan(x/c)^2/c + 4*arctan(c/x)*arctan(x/c)^3/c + arctan(x/c)^4/ 
c)*b^3*c^2 + 3*b^3*c^2*integrate(1/32*arctan(c/x)*log(c^2 + x^2)^2/(c^2 + 
x^2), x) + 12*b^3*c*integrate(1/32*x*arctan(c/x)^2/(c^2 + x^2), x) - 3*b^3 
*c*integrate(1/32*x*log(c^2 + x^2)^2/(c^2 + x^2), x) + 3/2*(2*x*arctan(c/x 
) + c*log(c^2 + x^2))*a^2*b + a^3*x + 28*b^3*integrate(1/32*x^2*arctan(c/x 
)^3/(c^2 + x^2), x) + 3*b^3*integrate(1/32*x^2*arctan(c/x)*log(c^2 + x^2)^ 
2/(c^2 + x^2), x) + 96*a*b^2*integrate(1/32*x^2*arctan(c/x)^2/(c^2 + x^2), 
 x) + 12*b^3*integrate(1/32*x^2*arctan(c/x)*log(c^2 + x^2)/(c^2 + x^2), x)
 

3.2.50.8 Giac [F]

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

input

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

output

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

3.2.50.9 Mupad [F(-1)]

Timed out. \[ \int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]

input

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

output

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