3.2.26 \(\int \frac {(a+b \arctan (c x^3))^3}{x^7} \, dx\) [126]

3.2.26.1 Optimal result

Integrand size = 16, antiderivative size = 146 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=-\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+b^2 c^2 \left (a+b \arctan \left (c x^3\right )\right ) \log \left (2-\frac {2}{1-i c x^3}\right )-\frac {1}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x^3}\right ) \]

output

-1/2*I*b*c^2*(a+b*arctan(c*x^3))^2-1/2*b*c*(a+b*arctan(c*x^3))^2/x^3-1/6*c 
^2*(a+b*arctan(c*x^3))^3-1/6*(a+b*arctan(c*x^3))^3/x^6+b^2*c^2*(a+b*arctan 
(c*x^3))*ln(2-2/(1-I*c*x^3))-1/2*I*b^3*c^2*polylog(2,-1+2/(1-I*c*x^3))
 

3.2.26.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=-\frac {3 b^2 \left (a+a c^2 x^6+b c x^3 \left (1+i c x^3\right )\right ) \arctan \left (c x^3\right )^2+b^3 \left (1+c^2 x^6\right ) \arctan \left (c x^3\right )^3+3 b \arctan \left (c x^3\right ) \left (a \left (a+2 b c x^3+a c^2 x^6\right )-2 b^2 c^2 x^6 \log \left (1-e^{2 i \arctan \left (c x^3\right )}\right )\right )+a \left (a \left (a+3 b c x^3\right )-6 b^2 c^2 x^6 \log \left (\frac {c x^3}{\sqrt {1+c^2 x^6}}\right )\right )+3 i b^3 c^2 x^6 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (c x^3\right )}\right )}{6 x^6} \]

input

Integrate[(a + b*ArcTan[c*x^3])^3/x^7,x]
 

output

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

3.2.26.3 Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5363, 5361, 5453, 5361, 5419, 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.

\(\displaystyle \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx\)

\(\Big \downarrow \) 5363

\(\displaystyle \frac {1}{3} \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^9}dx^3\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 2897

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

input

Int[(a + b*ArcTan[c*x^3])^3/x^7,x]
 

output

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

3.2.26.3.1 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 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]
 

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]
 

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

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

Time = 4.88 (sec) , antiderivative size = 11581, normalized size of antiderivative = 79.32

method	result	size
default	\(\text {Expression too large to display}\)	\(11581\)
parts	\(\text {Expression too large to display}\)	\(11581\)

input

int((a+b*arctan(c*x^3))^3/x^7,x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.2.26.5 Fricas [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]

input

integrate((a+b*arctan(c*x^3))^3/x^7,x, algorithm="fricas")
 

output

integral((b^3*arctan(c*x^3)^3 + 3*a*b^2*arctan(c*x^3)^2 + 3*a^2*b*arctan(c 
*x^3) + a^3)/x^7, x)
 

3.2.26.6 Sympy [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{3} \right )}\right )^{3}}{x^{7}}\, dx \]

input

integrate((a+b*atan(c*x**3))**3/x**7,x)
 

output

Integral((a + b*atan(c*x**3))**3/x**7, x)
 

3.2.26.7 Maxima [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]

input

integrate((a+b*arctan(c*x^3))^3/x^7,x, algorithm="maxima")
 

output

-1/2*((c*arctan(c*x^3) + 1/x^3)*c + arctan(c*x^3)/x^6)*a^2*b + 1/2*((arcta 
n(c*x^3)^2 - log(c^2*x^6 + 1) + 6*log(x))*c^2 - 2*(c*arctan(c*x^3) + 1/x^3 
)*c*arctan(c*x^3))*a*b^2 - 1/2*a*b^2*arctan(c*x^3)^2/x^6 + 1/192*(192*x^6* 
integrate(-1/64*(12*c^2*x^6*arctan(c*x^3)*log(c^2*x^6 + 1) - 12*c*x^3*arct 
an(c*x^3)^2 - 56*(c^2*x^6 + 1)*arctan(c*x^3)^3 + 3*(c*x^3 - 2*(c^2*x^6 + 1 
)*arctan(c*x^3))*log(c^2*x^6 + 1)^2)/(c^2*x^13 + x^7), x) - 4*arctan(c*x^3 
)^3 + 3*arctan(c*x^3)*log(c^2*x^6 + 1)^2)*b^3/x^6 - 1/6*a^3/x^6
 

3.2.26.8 Giac [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]

input

integrate((a+b*arctan(c*x^3))^3/x^7,x, algorithm="giac")
 

output

integrate((b*arctan(c*x^3) + a)^3/x^7, x)
 

3.2.26.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^3}{x^7} \,d x \]

input

int((a + b*atan(c*x^3))^3/x^7,x)
 

output

int((a + b*atan(c*x^3))^3/x^7, x)