Integrand size = 10, antiderivative size = 74 \[ \int x^2 \cot (a+b x) \, dx=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \] Output:
-1/3*I*x^3+x^2*ln(1-exp(2*I*(b*x+a)))/b-I*x*polylog(2,exp(2*I*(b*x+a)))/b^ 2+1/2*polylog(3,exp(2*I*(b*x+a)))/b^3
Time = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84 \[ \int x^2 \cot (a+b x) \, dx=\frac {i b^3 x^3+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )}{3 b^3} \] Input:
Integrate[x^2*Cot[a + b*x],x]
Output:
(I*b^3*x^3 + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^( (-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*P olyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*Poly Log[3, E^((-I)*(a + b*x))])/(3*b^3)
Time = 0.50 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.46, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 25, 4202, 2620, 3011, 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.
\(\displaystyle \int x^2 \cot (a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -x^2 \tan \left (a+b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int x^2 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle 2 i \int \frac {e^{i (2 a+2 b x+\pi )} x^2}{1+e^{i (2 a+2 b x+\pi )}}dx-\frac {i x^3}{3}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (\frac {i \int x \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i x^3}{3}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle 2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i \int \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i x^3}{3}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle 2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i x^3}{3}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle 2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i x^3}{3}\) |
Input:
Int[x^2*Cot[a + b*x],x]
Output:
(-1/3*I)*x^3 + (2*I)*(((-1/2*I)*x^2*Log[1 + E^(I*(2*a + Pi + 2*b*x))])/b + (I*(((I/2)*x*PolyLog[2, -E^(I*(2*a + Pi + 2*b*x))])/b - PolyLog[3, -E^(I* (2*a + Pi + 2*b*x))]/(4*b^2)))/b)
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (62 ) = 124\).
Time = 0.34 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.68
method | result | size |
risch | \(-\frac {i x^{3}}{3}+\frac {4 i a^{3}}{3 b^{3}}+\frac {2 i a^{2} x}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {a^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {2 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\) | \(198\) |
Input:
int(x^2*cot(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/3*I*x^3+4/3*I/b^3*a^3+2*I/b^2*a^2*x+1/b*ln(1-exp(I*(b*x+a)))*x^2-1/b^3* a^2*ln(1-exp(I*(b*x+a)))-2*I/b^2*polylog(2,-exp(I*(b*x+a)))*x+2/b^3*polylo g(3,exp(I*(b*x+a)))+1/b*ln(exp(I*(b*x+a))+1)*x^2-2*I/b^2*polylog(2,exp(I*( b*x+a)))*x+2/b^3*polylog(3,-exp(I*(b*x+a)))+1/b^3*a^2*ln(exp(I*(b*x+a))-1) -2/b^3*a^2*ln(exp(I*(b*x+a)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (59) = 118\).
Time = 0.10 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.30 \[ \int x^2 \cot (a+b x) \, dx=\frac {-2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \, b^{3}} \] Input:
integrate(x^2*cot(b*x+a),x, algorithm="fricas")
Output:
1/4*(-2*I*b*x*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + 2*I*b*x*dilog (cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 2*a^2*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*a^2*log(-1/2*cos(2*b*x + 2*a) - 1/2*I* sin(2*b*x + 2*a) + 1/2) + 2*(b^2*x^2 - a^2)*log(-cos(2*b*x + 2*a) + I*sin( 2*b*x + 2*a) + 1) + 2*(b^2*x^2 - a^2)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1) + polylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + polylog( 3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)))/b^3
\[ \int x^2 \cot (a+b x) \, dx=\int x^{2} \cot {\left (a + b x \right )}\, dx \] Input:
integrate(x**2*cot(b*x+a),x)
Output:
Integral(x**2*cot(a + b*x), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (59) = 118\).
Time = 0.10 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.47 \[ \int x^2 \cot (a+b x) \, dx=-\frac {2 i \, {\left (b x + a\right )}^{3} - 6 i \, {\left (b x + a\right )}^{2} a + 12 i \, b x {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 12 i \, b x {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - 6 \, a^{2} \log \left (\sin \left (b x + a\right )\right ) + 6 \, {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) + 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 12 \, {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 12 \, {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )})}{6 \, b^{3}} \] Input:
integrate(x^2*cot(b*x+a),x, algorithm="maxima")
Output:
-1/6*(2*I*(b*x + a)^3 - 6*I*(b*x + a)^2*a + 12*I*b*x*dilog(-e^(I*b*x + I*a )) + 12*I*b*x*dilog(e^(I*b*x + I*a)) - 6*a^2*log(sin(b*x + a)) + 6*(-I*(b* x + a)^2 + 2*I*(b*x + a)*a)*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 6*(I *(b*x + a)^2 - 2*I*(b*x + a)*a)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 3*((b*x + a)^2 - 2*(b*x + a)*a)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*c os(b*x + a) + 1) - 3*((b*x + a)^2 - 2*(b*x + a)*a)*log(cos(b*x + a)^2 + si n(b*x + a)^2 - 2*cos(b*x + a) + 1) - 12*polylog(3, -e^(I*b*x + I*a)) - 12* polylog(3, e^(I*b*x + I*a)))/b^3
\[ \int x^2 \cot (a+b x) \, dx=\int { x^{2} \cot \left (b x + a\right ) \,d x } \] Input:
integrate(x^2*cot(b*x+a),x, algorithm="giac")
Output:
integrate(x^2*cot(b*x + a), x)
Timed out. \[ \int x^2 \cot (a+b x) \, dx=\int x^2\,\mathrm {cot}\left (a+b\,x\right ) \,d x \] Input:
int(x^2*cot(a + b*x),x)
Output:
int(x^2*cot(a + b*x), x)
\[ \int x^2 \cot (a+b x) \, dx=\int \cot \left (b x +a \right ) x^{2}d x \] Input:
int(x^2*cot(b*x+a),x)
Output:
int(cot(a + b*x)*x**2,x)