Integrand size = 14, antiderivative size = 138 \[ \int x^2 (a+b \arctan (c x))^2 \, dx=\frac {b^2 x}{3 c^2}-\frac {b^2 \arctan (c x)}{3 c^3}-\frac {b x^2 (a+b \arctan (c x))}{3 c}-\frac {i (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
1/3*b^2*x/c^2-1/3*b^2*arctan(c*x)/c^3-1/3*b*x^2*(a+b*arctan(c*x))/c-1/3*I* (a+b*arctan(c*x))^2/c^3+1/3*x^3*(a+b*arctan(c*x))^2-2/3*b*(a+b*arctan(c*x) )*ln(2/(1+I*c*x))/c^3-1/3*I*b^2*polylog(2,1-2/(1+I*c*x))/c^3
Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.95 \[ \int x^2 (a+b \arctan (c x))^2 \, dx=\frac {b^2 c x-a b c^2 x^2+a^2 c^3 x^3+b^2 \left (i+c^3 x^3\right ) \arctan (c x)^2-b \arctan (c x) \left (b+b c^2 x^2-2 a c^3 x^3+2 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+a b \log \left (1+c^2 x^2\right )+i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{3 c^3} \]
(b^2*c*x - a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(I + c^3*x^3)*ArcTan[c*x]^2 - b *ArcTan[c*x]*(b + b*c^2*x^2 - 2*a*c^3*x^3 + 2*b*Log[1 + E^((2*I)*ArcTan[c* x])]) + a*b*Log[1 + c^2*x^2] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/( 3*c^3)
Time = 0.70 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5361, 5451, 5361, 262, 216, 5455, 5379, 2849, 2752}
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.
| \(\displaystyle \int x^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}\right )\) |
(x^3*(a + b*ArcTan[c*x])^2)/3 - (2*b*c*(((x^2*(a + b*ArcTan[c*x]))/2 - (b* c*(x/c^2 - ArcTan[c*x]/c^3))/2)/c^2 - (((-1/2*I)*(a + b*ArcTan[c*x])^2)/(b *c^2) - (((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/c)/c)/c^2))/3
3.1.16.3.1 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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)]*(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_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*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]
Time = 1.83 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.72
| method | result | size |
| parts | \(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{3}}\) | \(238\) |
| derivativedivides | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b \left (\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{3}}\) | \(239\) |
| default | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b \left (\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{3}}\) | \(239\) |
| risch | \(\frac {i b^{2} \ln \left (-i c x +1\right )^{2}}{12 c^{3}}+\frac {b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{3}}{6}-\frac {2 i b^{2} \ln \left (c^{2} x^{2}+1\right )}{9 c^{3}}+\frac {5 i b^{2} \ln \left (-i c x +1\right )}{36 c^{3}}-\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}-\frac {i b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{6 c^{3}}+\frac {i b^{2} \ln \left (i c x +1\right ) x^{2}}{6 c}-\frac {i b^{2} \ln \left (-i c x +1\right ) x^{2}}{6 c}-\frac {b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} \ln \left (i c x +1\right )^{2} x^{3}}{12}-\frac {i b a \ln \left (i c x +1\right ) x^{3}}{3}+\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {b^{2} x}{3 c^{2}}-\frac {a b \,x^{2}}{3 c}-\frac {17 i b^{2}}{54 c^{3}}-\frac {b^{2} \arctan \left (c x \right )}{6 c^{3}}+\frac {a b \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}-\frac {11 a b}{9 c^{3}}-\frac {i a^{2}}{3 c^{3}}+\frac {i a b \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {a^{2} x^{3}}{3}-\frac {i b^{2} \ln \left (i c x +1\right )^{2}}{12 c^{3}}+\frac {11 i b^{2} \ln \left (i c x +1\right )}{36 c^{3}}\) | \(398\) |
1/3*a^2*x^3+b^2/c^3*(1/3*c^3*x^3*arctan(c*x)^2-1/3*c^2*x^2*arctan(c*x)+1/3 *arctan(c*x)*ln(c^2*x^2+1)+1/3*c*x-1/3*arctan(c*x)+1/6*I*(ln(c*x-I)*ln(c^2 *x^2+1)-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(c^2*x^2+1)-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))))+2*a*b/c^3*(1/3*c^3*x^3*arctan(c*x)-1/6*c^2*x^2+1 /6*ln(c^2*x^2+1))
\[ \int x^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
\[ \int x^2 (a+b \arctan (c x))^2 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}\, dx \]
\[ \int x^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/3*a^2*x^3 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4)) *a*b + 1/48*(4*x^3*arctan(c*x)^2 - x^3*log(c^2*x^2 + 1)^2 + 48*integrate(1 /48*(4*c^2*x^4*log(c^2*x^2 + 1) - 8*c*x^3*arctan(c*x) + 36*(c^2*x^4 + x^2) *arctan(c*x)^2 + 3*(c^2*x^4 + x^2)*log(c^2*x^2 + 1)^2)/(c^2*x^2 + 1), x))* b^2
\[ \int x^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2 \,d x \]