Integrand size = 16, antiderivative size = 152 \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{3} b^2 c^2 x+\frac {1}{3} b^2 c^3 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b c x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {2}{3} b c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )-\frac {1}{3} i b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]
1/3*b^2*c^2*x+1/3*b^2*c^3*arccot(x/c)+1/3*b*c*x^2*(a+b*arccot(x/c))-1/3*I* c^3*(a+b*arccot(x/c))^2+1/3*x^3*(a+b*arccot(x/c))^2+2/3*b*c^3*(a+b*arccot( x/c))*ln(2-2/(1-I*c/x))-1/3*I*b^2*c^3*polylog(2,-1+2/(1-I*c/x))
Time = 0.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{3} \left (b^2 c^2 x+a b c x^2+a^2 x^3+b^2 \left (-i c^3+x^3\right ) \arctan \left (\frac {c}{x}\right )^2+b \arctan \left (\frac {c}{x}\right ) \left (2 a x^3+b c \left (c^2+x^2\right )+2 b c^3 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )-a b c^3 \log \left (1+\frac {c^2}{x^2}\right )+2 a b c^3 \log \left (\frac {c}{x}\right )-i b^2 c^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \]
(b^2*c^2*x + a*b*c*x^2 + a^2*x^3 + b^2*((-I)*c^3 + x^3)*ArcTan[c/x]^2 + b* ArcTan[c/x]*(2*a*x^3 + b*c*(c^2 + x^2) + 2*b*c^3*Log[1 - E^((2*I)*ArcTan[c /x])]) - a*b*c^3*Log[1 + c^2/x^2] + 2*a*b*c^3*Log[c/x] - I*b^2*c^3*PolyLog [2, E^((2*I)*ArcTan[c/x])])/3
Time = 0.76 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5363, 5361, 5453, 5361, 264, 216, 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.
| \(\displaystyle \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle -\int x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (\int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )d\frac {1}{x}-c^2 \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \int \frac {x^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-x\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (-\left (c^2 \left (i \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c}{x}+i}d\frac {1}{x}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (-\left (c^2 \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-\frac {i c}{x}}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2-\frac {2}{3} b c \left (-\left (c^2 \left (i \left (-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )\) |
(x^3*(a + b*ArcTan[c/x])^2)/3 - (2*b*c*(-1/2*(x^2*(a + b*ArcTan[c/x])) + ( b*c*(-x - c*ArcTan[c/x]))/2 - c^2*(((-1/2*I)*(a + b*ArcTan[c/x])^2)/b + I* ((-I)*(a + b*ArcTan[c/x])*Log[2 - 2/(1 - (I*c)/x)] - (b*PolyLog[2, -1 + 2/ (1 - (I*c)/x)])/2))))/3
3.2.41.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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]]
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]
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_.)/((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]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (134 ) = 268\).
Time = 3.78 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.36
| method | result | size |
| derivativedivides | \(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}+b^{2} \left (-\frac {x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{3 c^{3}}+\frac {\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{3 c^{2}}-\frac {2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )}{3}+\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (\frac {c}{x}\right )}{3}-\frac {x}{3 c}-\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{3}+\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{3}-\frac {i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{3}+\frac {i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{3 c^{3}}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {x^{2}}{6 c^{2}}-\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\) | \(358\) |
| default | \(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}+b^{2} \left (-\frac {x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{3 c^{3}}+\frac {\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{3 c^{2}}-\frac {2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )}{3}+\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (\frac {c}{x}\right )}{3}-\frac {x}{3 c}-\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{3}+\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{3}-\frac {i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{3}+\frac {i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{3 c^{3}}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {x^{2}}{6 c^{2}}-\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\) | \(358\) |
| parts | \(\frac {a^{2} x^{3}}{3}+\frac {b^{2} x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{3}+\frac {c \,b^{2} x^{2} \arctan \left (\frac {c}{x}\right )}{3}+\frac {2 c^{3} b^{2} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )}{3}-\frac {c^{3} b^{2} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}+i\right )^{2}}{12}+\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{3}+\frac {i c^{3} b^{2} \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{3}-\frac {i c^{3} b^{2} \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{3}+\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}-i\right )^{2}}{12}-\frac {i c^{3} b^{2} \operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}+\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{3}-\frac {b^{2} c^{3} \arctan \left (\frac {x}{c}\right )}{3}+\frac {b^{2} c^{2} x}{3}-\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}+\frac {i c^{3} b^{2} \operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}+\frac {i c^{3} b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-2 a b \,c^{3} \left (-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{3 c^{3}}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {x^{2}}{6 c^{2}}-\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\) | \(444\) |
| risch | \(\text {Expression too large to display}\) | \(25072\) |
-c^3*(-1/3*a^2/c^3*x^3+b^2*(-1/3/c^3*x^3*arctan(c/x)^2+1/3*arctan(c/x)*ln( 1+c^2/x^2)-1/3/c^2*x^2*arctan(c/x)-2/3*ln(c/x)*arctan(c/x)+1/6*I*(ln(c/x-I )*ln(1+c^2/x^2)-1/2*ln(c/x-I)^2-dilog(-1/2*I*(c/x+I))-ln(c/x-I)*ln(-1/2*I* (c/x+I)))-1/6*I*(ln(c/x+I)*ln(1+c^2/x^2)-1/2*ln(c/x+I)^2-dilog(1/2*I*(c/x- I))-ln(c/x+I)*ln(1/2*I*(c/x-I)))-1/3*arctan(c/x)-1/3*x/c-1/3*I*ln(c/x)*ln( 1+I*c/x)+1/3*I*ln(c/x)*ln(1-I*c/x)-1/3*I*dilog(1+I*c/x)+1/3*I*dilog(1-I*c/ x))+2*a*b*(-1/3/c^3*x^3*arctan(c/x)+1/6*ln(1+c^2/x^2)-1/6/c^2*x^2-1/3*ln(c /x)))
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{2} x^{2} \,d x } \]
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{2}\, dx \]
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{2} x^{2} \,d x } \]
1/3*a^2*x^3 + 1/3*(2*x^3*arctan(c/x) - (c^2*log(c^2 + x^2) - x^2)*c)*a*b + 1/48*(4*x^3*arctan2(c, x)^2 - x^3*log(c^2 + x^2)^2 + 48*integrate(1/48*(3 6*c^2*x^2*arctan2(c, x)^2 + 36*x^4*arctan2(c, x)^2 + 8*c*x^3*arctan2(c, x) + 4*x^4*log(c^2 + x^2) + 3*(c^2*x^2 + x^4)*log(c^2 + x^2)^2)/(c^2 + x^2), x))*b^2
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^2 \,d x \]