\(\int \frac {(a+b \arctan (c x^2))^3}{x^3} \, dx\) [89]

Optimal result

Integrand size = 16, antiderivative size = 138 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^3} \, dx=-\frac {1}{2} i c \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{2 x^2}+\frac {3}{2} b c \left (a+b \arctan \left (c x^2\right )\right )^2 \log \left (2-\frac {2}{1-i c x^2}\right )-\frac {3}{2} i b^2 c \left (a+b \arctan \left (c x^2\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x^2}\right )+\frac {3}{4} b^3 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i c x^2}\right ) \] Output:

-1/2*I*c*(a+b*arctan(c*x^2))^3-1/2*(a+b*arctan(c*x^2))^3/x^2+3/2*b*c*(a+b* 
arctan(c*x^2))^2*ln(2-2/(1-I*c*x^2))-3/2*I*b^2*c*(a+b*arctan(c*x^2))*polyl 
og(2,-1+2/(1-I*c*x^2))+3/4*b^3*c*polylog(3,-1+2/(1-I*c*x^2))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a^3}{x^2}-\frac {6 a^2 b \arctan \left (c x^2\right )}{x^2}+12 a^2 b c \log (x)-3 a^2 b c \log \left (1+c^2 x^4\right )+6 a b^2 c \left (\arctan \left (c x^2\right ) \left (\left (-i-\frac {1}{c x^2}\right ) \arctan \left (c x^2\right )+2 \log \left (1-e^{2 i \arctan \left (c x^2\right )}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (c x^2\right )}\right )\right )+2 b^3 c \left (-\frac {i \pi ^3}{8}+i \arctan \left (c x^2\right )^3-\frac {\arctan \left (c x^2\right )^3}{c x^2}+3 \arctan \left (c x^2\right )^2 \log \left (1-e^{-2 i \arctan \left (c x^2\right )}\right )+3 i \arctan \left (c x^2\right ) \operatorname {PolyLog}\left (2,e^{-2 i \arctan \left (c x^2\right )}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan \left (c x^2\right )}\right )\right )\right ) \] Input:

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

Output:

((-2*a^3)/x^2 - (6*a^2*b*ArcTan[c*x^2])/x^2 + 12*a^2*b*c*Log[x] - 3*a^2*b* 
c*Log[1 + c^2*x^4] + 6*a*b^2*c*(ArcTan[c*x^2]*((-I - 1/(c*x^2))*ArcTan[c*x 
^2] + 2*Log[1 - E^((2*I)*ArcTan[c*x^2])]) - I*PolyLog[2, E^((2*I)*ArcTan[c 
*x^2])]) + 2*b^3*c*((-1/8*I)*Pi^3 + I*ArcTan[c*x^2]^3 - ArcTan[c*x^2]^3/(c 
*x^2) + 3*ArcTan[c*x^2]^2*Log[1 - E^((-2*I)*ArcTan[c*x^2])] + (3*I)*ArcTan 
[c*x^2]*PolyLog[2, E^((-2*I)*ArcTan[c*x^2])] + (3*PolyLog[3, E^((-2*I)*Arc 
Tan[c*x^2])])/2))/4
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5363, 5361, 5459, 5403, 5527, 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.

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

\(\Big \downarrow \) 5363

\(\displaystyle \frac {1}{2} \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^4}dx^2\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

\(\displaystyle \frac {1}{2} \left (-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^2}+3 b c \left (i \left (2 i b c \int \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log \left (2-\frac {2}{1-i c x^2}\right )}{c^2 x^4+1}dx^2-i \log \left (2-\frac {2}{1-i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )^2\right )-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^3}{3 b}\right )\right )\)

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

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 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 5459

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

rule 5527

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*p*(I/2)   Int[(a + b*ArcTan[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]
 

Maple [F]

\[\int \frac {{\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{3}}{x^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-3/4*(c*(log(c^2*x^4 + 1) - log(x^4)) + 2*arctan(c*x^2)/x^2)*a^2*b - 1/2*a 
^3/x^2 - 1/64*(4*b^3*arctan(c*x^2)^3 - 3*b^3*arctan(c*x^2)*log(c^2*x^4 + 1 
)^2 - 64*x^2*integrate(-1/32*(12*b^3*c^2*x^4*arctan(c*x^2)*log(c^2*x^4 + 1 
) - 28*(b^3*c^2*x^4 + b^3)*arctan(c*x^2)^3 - 12*(8*a*b^2*c^2*x^4 + b^3*c*x 
^2 + 8*a*b^2)*arctan(c*x^2)^2 + 3*(b^3*c*x^2 - (b^3*c^2*x^4 + b^3)*arctan( 
c*x^2))*log(c^2*x^4 + 1)^2)/(c^2*x^7 + x^3), x))/x^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^3}{x^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{x^3} \, dx=\frac {-2 \mathit {atan} \left (c \,x^{2}\right )^{3} b^{3}-6 \mathit {atan} \left (c \,x^{2}\right )^{2} a \,b^{2}-6 \mathit {atan} \left (c \,x^{2}\right ) a^{2} b +24 \left (\int \frac {\mathit {atan} \left (c \,x^{2}\right )}{c^{2} x^{5}+x}d x \right ) a \,b^{2} c \,x^{2}+12 \left (\int \frac {\mathit {atan} \left (c \,x^{2}\right )^{2}}{c^{2} x^{5}+x}d x \right ) b^{3} c \,x^{2}-3 \,\mathrm {log}\left (-\sqrt {c}\, \sqrt {2}\, x +c \,x^{2}+1\right ) a^{2} b c \,x^{2}-3 \,\mathrm {log}\left (\sqrt {c}\, \sqrt {2}\, x +c \,x^{2}+1\right ) a^{2} b c \,x^{2}+12 \,\mathrm {log}\left (x \right ) a^{2} b c \,x^{2}-2 a^{3}}{4 x^{2}} \] Input:

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

Output:

( - 2*atan(c*x**2)**3*b**3 - 6*atan(c*x**2)**2*a*b**2 - 6*atan(c*x**2)*a** 
2*b + 24*int(atan(c*x**2)/(c**2*x**5 + x),x)*a*b**2*c*x**2 + 12*int(atan(c 
*x**2)**2/(c**2*x**5 + x),x)*b**3*c*x**2 - 3*log( - sqrt(c)*sqrt(2)*x + c* 
x**2 + 1)*a**2*b*c*x**2 - 3*log(sqrt(c)*sqrt(2)*x + c*x**2 + 1)*a**2*b*c*x 
**2 + 12*log(x)*a**2*b*c*x**2 - 2*a**3)/(4*x**2)