Integrand size = 21, antiderivative size = 154 \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {b x^4}{12}+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {\operatorname {PolyLog}\left (4,i c e^{2 i a+2 i b x}\right )}{8 b^3} \]
1/12*b*x^4-1/3*x^3*arctan(-c-(1-I*c)*cot(b*x+a))+1/6*I*x^3*ln(1-I*c*exp(2* I*a+2*I*b*x))+1/4*x^2*polylog(2,I*c*exp(2*I*a+2*I*b*x))/b+1/4*I*x*polylog( 3,I*c*exp(2*I*a+2*I*b*x))/b^2-1/8*polylog(4,I*c*exp(2*I*a+2*I*b*x))/b^3
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {1}{24} \left (8 x^3 \arctan (c+(1-i c) \cot (a+b x))+4 i x^3 \log \left (1+\frac {i e^{-2 i (a+b x)}}{c}\right )-\frac {6 x^2 \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{c}\right )}{b}+\frac {6 i x \operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{c}\right )}{b^2}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {i e^{-2 i (a+b x)}}{c}\right )}{b^3}\right ) \]
(8*x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (4*I)*x^3*Log[1 + I/(c*E^((2*I )*(a + b*x)))] - (6*x^2*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))])/b + ((6* I)*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, (-I)/(c *E^((2*I)*(a + b*x)))])/b^3)/24
Time = 0.80 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.29, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5696, 25, 2615, 2620, 3011, 7163, 2720, 7143}
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.
\(\displaystyle \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx\) |
\(\Big \downarrow \) 5696 |
\(\displaystyle \frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))-\frac {1}{3} i b \int -\frac {x^3}{e^{2 i a+2 i b x} c+i}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} i b \int \frac {x^3}{e^{2 i a+2 i b x} c+i}dx+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {1}{3} i b \left (i c \int \frac {e^{2 i a+2 i b x} x^3}{e^{2 i a+2 i b x} c+i}dx-\frac {i x^4}{4}\right )+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{3} i b \left (i c \left (\frac {3 i \int x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )dx}{2 b c}-\frac {i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^4}{4}\right )+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{3} i b \left (i c \left (\frac {3 i \left (\frac {i x^2 \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \int x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )dx}{b}\right )}{2 b c}-\frac {i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^4}{4}\right )+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {1}{3} i b \left (i c \left (\frac {3 i \left (\frac {i x^2 \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \left (\frac {i \int \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )dx}{2 b}-\frac {i x \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{2 b}\right )}{b}\right )}{2 b c}-\frac {i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^4}{4}\right )+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{3} i b \left (i c \left (\frac {3 i \left (\frac {i x^2 \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \left (\frac {\int e^{-2 i a-2 i b x} \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2}-\frac {i x \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{2 b}\right )}{b}\right )}{2 b c}-\frac {i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^4}{4}\right )+\frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{3} x^3 \arctan (c+(1-i c) \cot (a+b x))+\frac {1}{3} i b \left (i c \left (\frac {3 i \left (\frac {i x^2 \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \left (\frac {\operatorname {PolyLog}\left (4,i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {i x \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{2 b}\right )}{b}\right )}{2 b c}-\frac {i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^4}{4}\right )\) |
(x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/3 + (I/3)*b*((-1/4*I)*x^4 + I*c*( ((-1/2*I)*x^3*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)])/(b*c) + (((3*I)/2)*((( I/2)*x^2*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/b - (I*(((-1/2*I)*x*Poly Log[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b + PolyLog[4, I*c*E^((2*I)*a + (2*I) *b*x)]/(4*b^2)))/b))/(b*c)))
3.1.65.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_. ), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1))), x] - Simp[I*(b/(f*(m + 1))) Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2 *I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && E qQ[(c - I*d)^2, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.42 (sec) , antiderivative size = 1487, normalized size of antiderivative = 9.66
-1/12*I*(2*I*Pi-2*ln(I+c)+I*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I+c))* csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))+I*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*(I +c)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))- 1))+I*Pi*csgn(I*(exp(2*I*(b*x+a))*c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp (2*I*(b*x+a))-1))^2+I*Pi*csgn(I*exp(2*I*(b*x+a)))^3-I*Pi*csgn(I*(exp(2*I*( b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b *x+a))-1))-I*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(e xp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2+I*Pi*csgn(I/(exp(2*I*(b*x+a) )-1))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2+I*Pi*csgn((exp (2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^3-I*Pi*csgn((exp(2*I*(b*x+a))*c+I )/(exp(2*I*(b*x+a))-1))^2-I*Pi*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x +a))-1))^3+I*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(e xp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))-I*Pi*csgn(I*(I+c))*csgn(I*(I+c )/(exp(2*I*(b*x+a))-1))^2-I*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b* x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2-I*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn (I*(I+c)/(exp(2*I*(b*x+a))-1))^2-I*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I* (exp(2*I*(b*x+a))*c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1) )-I*Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(e xp(2*I*(b*x+a))-1))^2+I*Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a) ))+I*Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^3+I*Pi*csgn(I*exp(2*I*(b*x+a...
Time = 0.26 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08 \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {2 \, b^{4} x^{4} + 4 i \, b^{3} x^{3} \log \left (-\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 2 \, a^{4} - 4 i \, a^{3} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + 6 i \, b x {\rm polylog}\left (3, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 4 \, {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (4, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{3}} \]
1/24*(2*b^4*x^4 + 4*I*b^3*x^3*log(-(c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b *x + 2*I*a) + I)) + 6*b^2*x^2*dilog(I*c*e^(2*I*b*x + 2*I*a)) - 2*a^4 - 4*I *a^3*log((c*e^(2*I*b*x + 2*I*a) + I)/c) + 6*I*b*x*polylog(3, I*c*e^(2*I*b* x + 2*I*a)) - 4*(-I*b^3*x^3 - I*a^3)*log(-I*c*e^(2*I*b*x + 2*I*a) + 1) - 3 *polylog(4, I*c*e^(2*I*b*x + 2*I*a)))/b^3
Exception generated. \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
Exception raised: CoercionFailed >> Cannot convert _t0**2 - exp(2*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[x,b,_t0,exp(I*a)]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.01 \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {\frac {4 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \arctan \left ({\left (-i \, c + 1\right )} \cot \left (b x + a\right ) + c\right )}{b^{2}} + \frac {{\left (-3 i \, {\left (b x + a\right )}^{4} + 12 i \, {\left (b x + a\right )}^{3} a - 18 i \, {\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (4 i \, {\left (b x + a\right )}^{3} - 9 i \, {\left (b x + a\right )}^{2} a + 9 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - 3 \, {\left (4 i \, {\left (b x + a\right )}^{2} - 6 i \, {\left (b x + a\right )} a + 3 i \, a^{2}\right )} {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 9 \, {\left (b x + a\right )} a^{2}\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left (4 \, b x + a\right )} {\rm Li}_{3}(i \, c e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \, {\rm Li}_{4}(i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c - 1\right )}}{b^{2} {\left (c + i\right )}}}{12 \, b} \]
1/12*(4*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*arctan((-I*c + 1 )*cot(b*x + a) + c)/b^2 + (-3*I*(b*x + a)^4 + 12*I*(b*x + a)^3*a - 18*I*(b *x + a)^2*a^2 - 2*(4*I*(b*x + a)^3 - 9*I*(b*x + a)^2*a + 9*I*(b*x + a)*a^2 )*arctan2(c*cos(2*b*x + 2*a), c*sin(2*b*x + 2*a) + 1) - 3*(4*I*(b*x + a)^2 - 6*I*(b*x + a)*a + 3*I*a^2)*dilog(I*c*e^(2*I*b*x + 2*I*a)) + (4*(b*x + a )^3 - 9*(b*x + a)^2*a + 9*(b*x + a)*a^2)*log(c^2*cos(2*b*x + 2*a)^2 + c^2* sin(2*b*x + 2*a)^2 + 2*c*sin(2*b*x + 2*a) + 1) + 3*(4*b*x + a)*polylog(3, I*c*e^(2*I*b*x + 2*I*a)) + 6*I*polylog(4, I*c*e^(2*I*b*x + 2*I*a)))*(I*c - 1)/(b^2*(c + I)))/b
\[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\int { -x^{2} \arctan \left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right ) \,d x } \]
Timed out. \[ \int x^2 \arctan (c+(1-i c) \cot (a+b x)) \, dx=\int x^2\,\mathrm {atan}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right ) \,d x \]