Integrand size = 16, antiderivative size = 136 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=-\frac {i \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{c}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{x}-\frac {3 b \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2}{1+\frac {i c}{x}}\right )}{c}-\frac {3 i b^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {i c}{x}}\right )}{c}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+\frac {i c}{x}}\right )}{2 c} \] Output:
-I*(a+b*arccot(x/c))^3/c-(a+b*arccot(x/c))^3/x-3*b*(a+b*arccot(x/c))^2*ln( 2/(1+I*c/x))/c-3*I*b^2*(a+b*arccot(x/c))*polylog(2,1-2/(1+I*c/x))/c-3/2*b^ 3*polylog(3,1-2/(1+I*c/x))/c
Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=-\frac {2 a^3 c+6 a^2 b c \arctan \left (\frac {c}{x}\right )+6 a b^2 c \arctan \left (\frac {c}{x}\right )^2-6 i a b^2 x \arctan \left (\frac {c}{x}\right )^2+2 b^3 c \arctan \left (\frac {c}{x}\right )^3-2 i b^3 x \arctan \left (\frac {c}{x}\right )^3+12 a b^2 x \arctan \left (\frac {c}{x}\right ) \log \left (1+e^{2 i \arctan \left (\frac {c}{x}\right )}\right )+6 b^3 x \arctan \left (\frac {c}{x}\right )^2 \log \left (1+e^{2 i \arctan \left (\frac {c}{x}\right )}\right )-3 a^2 b x \log \left (1+\frac {c^2}{x^2}\right )-6 i b^2 x \left (a+b \arctan \left (\frac {c}{x}\right )\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )+3 b^3 x \operatorname {PolyLog}\left (3,-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )}{2 c x} \] Input:
Integrate[(a + b*ArcTan[c/x])^3/x^2,x]
Output:
-1/2*(2*a^3*c + 6*a^2*b*c*ArcTan[c/x] + 6*a*b^2*c*ArcTan[c/x]^2 - (6*I)*a* b^2*x*ArcTan[c/x]^2 + 2*b^3*c*ArcTan[c/x]^3 - (2*I)*b^3*x*ArcTan[c/x]^3 + 12*a*b^2*x*ArcTan[c/x]*Log[1 + E^((2*I)*ArcTan[c/x])] + 6*b^3*x*ArcTan[c/x ]^2*Log[1 + E^((2*I)*ArcTan[c/x])] - 3*a^2*b*x*Log[1 + c^2/x^2] - (6*I)*b^ 2*x*(a + b*ArcTan[c/x])*PolyLog[2, -E^((2*I)*ArcTan[c/x])] + 3*b^3*x*PolyL og[3, -E^((2*I)*ArcTan[c/x])])/(c*x)
Time = 0.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5363, 5345, 5455, 5379, 5529, 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.
| \(\displaystyle \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle -\int \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle 3 b c \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\left (\frac {c^2}{x^2}+1\right ) x}d\frac {1}{x}-\frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle -\frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x}+3 b c \left (-\frac {\int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{i-\frac {c}{x}}d\frac {1}{x}}{c}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle -\frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x}+3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{c}-2 b \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right ) \log \left (\frac {2}{\frac {i c}{x}+1}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}}{c}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle -\frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x}+3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{c}-2 b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\frac {i c}{x}+1}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{\frac {i c}{x}+1}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{2 c}\right )}{c}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x}+3 b c \left (-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b c^2}-\frac {\frac {\log \left (\frac {2}{1+\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{c}-2 b \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{\frac {i c}{x}+1}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{\frac {i c}{x}+1}\right )}{4 c}\right )}{c}\right )\) |
Input:
Int[(a + b*ArcTan[c/x])^3/x^2,x]
Output:
-((a + b*ArcTan[c/x])^3/x) + 3*b*c*(((-1/3*I)*(a + b*ArcTan[c/x])^3)/(b*c^ 2) - (((a + b*ArcTan[c/x])^2*Log[2/(1 + (I*c)/x)])/c - 2*b*(((-1/2*I)*(a + b*ArcTan[c/x])*PolyLog[2, 1 - 2/(1 + (I*c)/x)])/c - (b*PolyLog[3, 1 - 2/( 1 + (I*c)/x)])/(4*c)))/c)
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[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])
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]]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[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 ]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (129 ) = 258\).
Time = 1.95 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.02
| method | result | size |
| derivativedivides | \(-\frac {\frac {c \,a^{3}}{x}+b^{3} \left (\arctan \left (\frac {c}{x}\right )^{3} \left (\frac {c}{x}+i\right )-2 i \arctan \left (\frac {c}{x}\right )^{3}+3 \arctan \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-3 i \arctan \left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )}{2}\right )+3 a \,b^{2} \left (\arctan \left (\frac {c}{x}\right )^{2} \left (\frac {c}{x}+i\right )+2 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-2 i \arctan \left (\frac {c}{x}\right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )\right )+3 a^{2} b \left (\frac {c \arctan \left (\frac {c}{x}\right )}{x}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}\right )}{c}\) | \(275\) |
| default | \(-\frac {\frac {c \,a^{3}}{x}+b^{3} \left (\arctan \left (\frac {c}{x}\right )^{3} \left (\frac {c}{x}+i\right )-2 i \arctan \left (\frac {c}{x}\right )^{3}+3 \arctan \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-3 i \arctan \left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )}{2}\right )+3 a \,b^{2} \left (\arctan \left (\frac {c}{x}\right )^{2} \left (\frac {c}{x}+i\right )+2 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-2 i \arctan \left (\frac {c}{x}\right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )\right )+3 a^{2} b \left (\frac {c \arctan \left (\frac {c}{x}\right )}{x}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}\right )}{c}\) | \(275\) |
| parts | \(-\frac {a^{3}}{x}-\frac {b^{3} \left (\arctan \left (\frac {c}{x}\right )^{3} \left (\frac {c}{x}+i\right )-2 i \arctan \left (\frac {c}{x}\right )^{3}+3 \arctan \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-3 i \arctan \left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )}{2}\right )}{c}-\frac {3 a^{2} b \arctan \left (\frac {c}{x}\right )}{x}+\frac {3 a^{2} b \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2 c}-\frac {3 a \,b^{2} \left (\arctan \left (\frac {c}{x}\right )^{2} \left (\frac {c}{x}+i\right )+2 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )-2 i \arctan \left (\frac {c}{x}\right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+\frac {i c}{x}\right )^{2}}{1+\frac {c^{2}}{x^{2}}}\right )\right )}{c}\) | \(281\) |
Input:
int((a+b*arctan(c/x))^3/x^2,x,method=_RETURNVERBOSE)
Output:
-1/c*(c/x*a^3+b^3*(arctan(c/x)^3*(c/x+I)-2*I*arctan(c/x)^3+3*arctan(c/x)^2 *ln(1+(1+I*c/x)^2/(1+c^2/x^2))-3*I*arctan(c/x)*polylog(2,-(1+I*c/x)^2/(1+c ^2/x^2))+3/2*polylog(3,-(1+I*c/x)^2/(1+c^2/x^2)))+3*a*b^2*(arctan(c/x)^2*( c/x+I)+2*arctan(c/x)*ln(1+(1+I*c/x)^2/(1+c^2/x^2))-2*I*arctan(c/x)^2-I*pol ylog(2,-(1+I*c/x)^2/(1+c^2/x^2)))+3*a^2*b*(c/x*arctan(c/x)-1/2*ln(1+c^2/x^ 2)))
\[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{2}} \,d x } \] Input:
integrate((a+b*arctan(c/x))^3/x^2,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^2, x)
\[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}}{x^{2}}\, dx \] Input:
integrate((a+b*atan(c/x))**3/x**2,x)
Output:
Integral((a + b*atan(c/x))**3/x**2, x)
\[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{2}} \,d x } \] Input:
integrate((a+b*arctan(c/x))^3/x^2,x, algorithm="maxima")
Output:
-3/2*a^2*b*(2*c*arctan(c/x)/x - log(c^2/x^2 + 1))/c - a^3/x - 1/32*(4*b^3* arctan2(c, x)^3 - 3*b^3*arctan2(c, x)*log(c^2 + x^2)^2 - (28*b^3*arctan(c/ x)^3*arctan(x/c)/c + 896*b^3*c^2*integrate(1/32*arctan(c/x)^3/(c^2*x^2 + x ^4), x) + 96*b^3*c^2*integrate(1/32*arctan(c/x)*log(c^2 + x^2)^2/(c^2*x^2 + x^4), x) + 3072*a*b^2*c^2*integrate(1/32*arctan(c/x)^2/(c^2*x^2 + x^4), x) + 96*a*b^2*arctan(c/x)^2*arctan(x/c)/c - 384*b^3*c*integrate(1/32*x*arc tan(c/x)^2/(c^2*x^2 + x^4), x) + 96*b^3*c*integrate(1/32*x*log(c^2 + x^2)^ 2/(c^2*x^2 + x^4), x) + 32*(3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/ c)*a*b^2 + 7*(6*arctan(c/x)^2*arctan(x/c)^2/c + 4*arctan(c/x)*arctan(x/c)^ 3/c + arctan(x/c)^4/c)*b^3 + 96*b^3*integrate(1/32*x^2*arctan(c/x)*log(c^2 + x^2)^2/(c^2*x^2 + x^4), x) - 384*b^3*integrate(1/32*x^2*arctan(c/x)*log (c^2 + x^2)/(c^2*x^2 + x^4), x))*x)/x
\[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{2}} \,d x } \] Input:
integrate((a+b*arctan(c/x))^3/x^2,x, algorithm="giac")
Output:
integrate((b*arctan(c/x) + a)^3/x^2, x)
Timed out. \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3}{x^2} \,d x \] Input:
int((a + b*atan(c/x))^3/x^2,x)
Output:
int((a + b*atan(c/x))^3/x^2, x)
\[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=\frac {-2 \mathit {atan} \left (\frac {c}{x}\right )^{3} b^{3} c -6 \mathit {atan} \left (\frac {c}{x}\right )^{2} a \,b^{2} c -6 \mathit {atan} \left (\frac {c}{x}\right ) a^{2} b c -12 \left (\int \frac {\mathit {atan} \left (\frac {c}{x}\right )}{c^{2} x +x^{3}}d x \right ) a \,b^{2} c^{2} x -6 \left (\int \frac {\mathit {atan} \left (\frac {c}{x}\right )^{2}}{c^{2} x +x^{3}}d x \right ) b^{3} c^{2} x +3 \,\mathrm {log}\left (c^{2}+x^{2}\right ) a^{2} b x -6 \,\mathrm {log}\left (x \right ) a^{2} b x -2 a^{3} c}{2 c x} \] Input:
int((a+b*atan(c/x))^3/x^2,x)
Output:
( - 2*atan(c/x)**3*b**3*c - 6*atan(c/x)**2*a*b**2*c - 6*atan(c/x)*a**2*b*c - 12*int(atan(c/x)/(c**2*x + x**3),x)*a*b**2*c**2*x - 6*int(atan(c/x)**2/ (c**2*x + x**3),x)*b**3*c**2*x + 3*log(c**2 + x**2)*a**2*b*x - 6*log(x)*a* *2*b*x - 2*a**3*c)/(2*c*x)