\(\int \frac {(a+b \arctan (c x^3))^2}{x^4} \, dx\) [118]

Optimal result

Integrand size = 16, antiderivative size = 100 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=-\frac {1}{3} i c \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{3 x^3}+\frac {2}{3} b c \left (a+b \arctan \left (c x^3\right )\right ) \log \left (2-\frac {2}{1-i c x^3}\right )-\frac {1}{3} i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x^3}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\frac {b^2 \left (-1-i c x^3\right ) \arctan \left (c x^3\right )^2+2 b \arctan \left (c x^3\right ) \left (-a+b c x^3 \log \left (1-e^{2 i \arctan \left (c x^3\right )}\right )\right )-a \left (a-2 b c x^3 \log \left (c x^3\right )+b c x^3 \log \left (1+c^2 x^6\right )\right )-i b^2 c x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (c x^3\right )}\right )}{3 x^3} \] Input:

Integrate[(a + b*ArcTan[c*x^3])^2/x^4,x]
 

Output:

(b^2*(-1 - I*c*x^3)*ArcTan[c*x^3]^2 + 2*b*ArcTan[c*x^3]*(-a + b*c*x^3*Log[ 
1 - E^((2*I)*ArcTan[c*x^3])]) - a*(a - 2*b*c*x^3*Log[c*x^3] + b*c*x^3*Log[ 
1 + c^2*x^6]) - I*b^2*c*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x^3])])/(3*x^3)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5363, 5361, 5459, 5403, 2897}

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.

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

\(\Big \downarrow \) 5363

\(\displaystyle \frac {1}{3} \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^6}dx^3\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

\(\displaystyle \frac {1}{3} \left (-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^3}+2 b c \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-i c x^3}\right )}{c^2 x^6+1}dx^3-i \log \left (2-\frac {2}{1-i c x^3}\right ) \left (a+b \arctan \left (c x^3\right )\right )\right )-\frac {i \left (a+b \arctan \left (c x^3\right )\right )^2}{2 b}\right )\right )\)

\(\Big \downarrow \) 2897

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

Input:

Int[(a + b*ArcTan[c*x^3])^2/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.57 (sec) , antiderivative size = 11455, normalized size of antiderivative = 114.55

method	result	size
default	\(\text {Expression too large to display}\)	\(11455\)
parts	\(\text {Expression too large to display}\)	\(11455\)

Input:

int((a+b*arctan(c*x^3))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctan(c*x^3))^2/x^4,x, algorithm="fricas")
 

Output:

integral((b^2*arctan(c*x^3)^2 + 2*a*b*arctan(c*x^3) + a^2)/x^4, x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{3} \right )}\right )^{2}}{x^{4}}\, dx \] Input:

integrate((a+b*atan(c*x**3))**2/x**4,x)
 

Output:

Integral((a + b*atan(c*x**3))**2/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctan(c*x^3))^2/x^4,x, algorithm="maxima")
 

Output:

-1/3*(c*(log(c^2*x^6 + 1) - log(x^6)) + 2*arctan(c*x^3)/x^3)*a*b + 1/48*(4 
8*x^3*integrate(-1/16*(4*c^2*x^6*log(c^2*x^6 + 1) - 8*c*x^3*arctan(c*x^3) 
- 12*(c^2*x^6 + 1)*arctan(c*x^3)^2 - (c^2*x^6 + 1)*log(c^2*x^6 + 1)^2)/(c^ 
2*x^10 + x^4), x) - 4*arctan(c*x^3)^2 + log(c^2*x^6 + 1)^2)*b^2/x^3 - 1/3* 
a^2/x^3
 

Giac [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctan(c*x^3))^2/x^4,x, algorithm="giac")
 

Output:

integrate((b*arctan(c*x^3) + a)^2/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^2}{x^4} \,d x \] Input:

int((a + b*atan(c*x^3))^2/x^4,x)
 

Output:

int((a + b*atan(c*x^3))^2/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^4} \, dx=\frac {-\mathit {atan} \left (c \,x^{3}\right )^{2} b^{2}-2 \mathit {atan} \left (c \,x^{3}\right ) a b +6 \left (\int \frac {\mathit {atan} \left (c \,x^{3}\right )}{c^{2} x^{7}+x}d x \right ) b^{2} c \,x^{3}-\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) a b c \,x^{3}-\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) a b c \,x^{3}-\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+1\right ) a b c \,x^{3}+6 \,\mathrm {log}\left (x \right ) a b c \,x^{3}-a^{2}}{3 x^{3}} \] Input:

int((a+b*atan(c*x^3))^2/x^4,x)
 

Output:

( - atan(c*x**3)**2*b**2 - 2*atan(c*x**3)*a*b + 6*int(atan(c*x**3)/(c**2*x 
**7 + x),x)*b**2*c*x**3 - log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)*x + 1)*a*b* 
c*x**3 - log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*a*b*c*x**3 - log(c**( 
2/3)*x**2 + 1)*a*b*c*x**3 + 6*log(x)*a*b*c*x**3 - a**2)/(3*x**3)