Integrand size = 10, antiderivative size = 91 \[ \int x \cot ^3(a+b x) \, dx=-\frac {x}{2 b}+\frac {i x^2}{2}-\frac {\cot (a+b x)}{2 b^2}-\frac {x \cot ^2(a+b x)}{2 b}-\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2} \] Output:
-1/2*x/b+1/2*I*x^2-1/2*cot(b*x+a)/b^2-1/2*x*cot(b*x+a)^2/b-x*ln(1-exp(2*I* (b*x+a)))/b+1/2*I*polylog(2,exp(2*I*(b*x+a)))/b^2
Time = 2.79 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int x \cot ^3(a+b x) \, dx=\frac {-i b \pi x-b^2 x^2 \cot (a)-b x \csc ^2(a+b x)-\pi \log \left (1+e^{-2 i b x}\right )-2 b x \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \left (i b x-\log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\log (\sin (b x+\arctan (\tan (a))))\right )+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )+b^2 e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}+\csc (a) \csc (a+b x) \sin (b x)}{2 b^2} \] Input:
Integrate[x*Cot[a + b*x]^3,x]
Output:
((-I)*b*Pi*x - b^2*x^2*Cot[a] - b*x*Csc[a + b*x]^2 - Pi*Log[1 + E^((-2*I)* b*x)] - 2*b*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*(I*b*x - Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + L og[Sin[b*x + ArcTan[Tan[a]]]]) + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a ]]))] + b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2] + Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^2)
Time = 0.51 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 25, 4203, 25, 3042, 25, 3954, 24, 4202, 2620, 2715, 2838}
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.
\(\displaystyle \int x \cot ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -x \tan \left (a+b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int x \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^3dx\) |
\(\Big \downarrow \) 4203 |
\(\displaystyle \frac {\int \cot ^2(a+b x)dx}{2 b}+\int -x \cot (a+b x)dx-\frac {x \cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot ^2(a+b x)dx}{2 b}-\int x \cot (a+b x)dx-\frac {x \cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -x \tan \left (a+b x+\frac {\pi }{2}\right )dx+\frac {\int \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{2 b}-\frac {x \cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{2 b}+\int x \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx-\frac {x \cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \int x \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx+\frac {-\int 1dx-\frac {\cot (a+b x)}{b}}{2 b}-\frac {x \cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \int x \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx-\frac {x \cot ^2(a+b x)}{2 b}+\frac {-\frac {\cot (a+b x)}{b}-x}{2 b}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle -2 i \int \frac {e^{i (2 a+2 b x+\pi )} x}{1+e^{i (2 a+2 b x+\pi )}}dx-\frac {x \cot ^2(a+b x)}{2 b}+\frac {-\frac {\cot (a+b x)}{b}-x}{2 b}+\frac {i x^2}{2}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -2 i \left (\frac {i \int \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i x \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {x \cot ^2(a+b x)}{2 b}+\frac {-\frac {\cot (a+b x)}{b}-x}{2 b}+\frac {i x^2}{2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -2 i \left (\frac {\int e^{-i (2 a+2 b x+\pi )} \log \left (1+e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}-\frac {i x \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {x \cot ^2(a+b x)}{2 b}+\frac {-\frac {\cot (a+b x)}{b}-x}{2 b}+\frac {i x^2}{2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -2 i \left (-\frac {\operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i x \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {x \cot ^2(a+b x)}{2 b}+\frac {-\frac {\cot (a+b x)}{b}-x}{2 b}+\frac {i x^2}{2}\) |
Input:
Int[x*Cot[a + b*x]^3,x]
Output:
(I/2)*x^2 - (x*Cot[a + b*x]^2)/(2*b) + (-x - Cot[a + b*x]/b)/(2*b) - (2*I) *(((-1/2*I)*x*Log[1 + E^(I*(2*a + Pi + 2*b*x))])/b - PolyLog[2, -E^(I*(2*a + Pi + 2*b*x))]/(4*b^2))
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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
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[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (75 ) = 150\).
Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.16
method | result | size |
risch | \(\frac {i x^{2}}{2}+\frac {2 b x \,{\mathrm e}^{2 i \left (b x +a \right )}-i {\mathrm e}^{2 i \left (b x +a \right )}+i}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {2 i a x}{b}+\frac {i a^{2}}{b^{2}}-\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}+\frac {i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {2 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}\) | \(197\) |
Input:
int(x*cot(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*I*x^2+(2*b*x*exp(2*I*(b*x+a))-I*exp(2*I*(b*x+a))+I)/b^2/(exp(2*I*(b*x+ a))-1)^2+2*I/b*a*x+I/b^2*a^2-1/b*ln(1-exp(I*(b*x+a)))*x-1/b^2*ln(1-exp(I*( b*x+a)))*a+I/b^2*polylog(2,exp(I*(b*x+a)))-1/b*ln(exp(I*(b*x+a))+1)*x+I/b^ 2*polylog(2,-exp(I*(b*x+a)))+1/b^2*a*ln(exp(I*(b*x+a))-1)-2/b^2*a*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. 291 vs. \(2 (72) = 144\).
Time = 0.09 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.20 \[ \int x \cot ^3(a+b x) \, dx=\frac {4 \, b x + {\left (i \, \cos \left (2 \, b x + 2 \, a\right ) - i\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + {\left (-i \, \cos \left (2 \, b x + 2 \, a\right ) + i\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, {\left (a \cos \left (2 \, b x + 2 \, a\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (a \cos \left (2 \, b x + 2 \, a\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b x - {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b x - {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, \sin \left (2 \, b x + 2 \, a\right )}{4 \, {\left (b^{2} \cos \left (2 \, b x + 2 \, a\right ) - b^{2}\right )}} \] Input:
integrate(x*cot(b*x+a)^3,x, algorithm="fricas")
Output:
1/4*(4*b*x + (I*cos(2*b*x + 2*a) - I)*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + (-I*cos(2*b*x + 2*a) + I)*dilog(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 2*(a*cos(2*b*x + 2*a) - a)*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin (2*b*x + 2*a) + 1/2) + 2*(a*cos(2*b*x + 2*a) - a)*log(-1/2*cos(2*b*x + 2*a ) - 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b*x - (b*x + a)*cos(2*b*x + 2*a) + a)*log(-cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a) + 1) + 2*(b*x - (b*x + a)*co s(2*b*x + 2*a) + a)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1) + 2*si n(2*b*x + 2*a))/(b^2*cos(2*b*x + 2*a) - b^2)
\[ \int x \cot ^3(a+b x) \, dx=\int x \cot ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(x*cot(b*x+a)**3,x)
Output:
Integral(x*cot(a + b*x)**3, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (72) = 144\).
Time = 0.16 (sec) , antiderivative size = 586, normalized size of antiderivative = 6.44 \[ \int x \cot ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(x*cot(b*x+a)^3,x, algorithm="maxima")
Output:
(b^2*x^2*cos(4*b*x + 4*a) + I*b^2*x^2*sin(4*b*x + 4*a) + b^2*x^2 - 2*(b*x* cos(4*b*x + 4*a) - 2*b*x*cos(2*b*x + 2*a) + I*b*x*sin(4*b*x + 4*a) - 2*I*b *x*sin(2*b*x + 2*a) + b*x)*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 2*(b* x*cos(4*b*x + 4*a) - 2*b*x*cos(2*b*x + 2*a) + I*b*x*sin(4*b*x + 4*a) - 2*I *b*x*sin(2*b*x + 2*a) + b*x)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 2* (b^2*x^2 + 2*I*b*x + 1)*cos(2*b*x + 2*a) + 2*(cos(4*b*x + 4*a) - 2*cos(2*b *x + 2*a) + I*sin(4*b*x + 4*a) - 2*I*sin(2*b*x + 2*a) + 1)*dilog(-e^(I*b*x + I*a)) + 2*(cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + I*sin(4*b*x + 4*a) - 2*I*sin(2*b*x + 2*a) + 1)*dilog(e^(I*b*x + I*a)) - (-I*b*x*cos(4*b*x + 4* a) + 2*I*b*x*cos(2*b*x + 2*a) + b*x*sin(4*b*x + 4*a) - 2*b*x*sin(2*b*x + 2 *a) - I*b*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - ( -I*b*x*cos(4*b*x + 4*a) + 2*I*b*x*cos(2*b*x + 2*a) + b*x*sin(4*b*x + 4*a) - 2*b*x*sin(2*b*x + 2*a) - I*b*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2* cos(b*x + a) + 1) + 2*(-I*b^2*x^2 + 2*b*x - I)*sin(2*b*x + 2*a) + 2)/(-2*I *b^2*cos(4*b*x + 4*a) + 4*I*b^2*cos(2*b*x + 2*a) + 2*b^2*sin(4*b*x + 4*a) - 4*b^2*sin(2*b*x + 2*a) - 2*I*b^2)
\[ \int x \cot ^3(a+b x) \, dx=\int { x \cot \left (b x + a\right )^{3} \,d x } \] Input:
integrate(x*cot(b*x+a)^3,x, algorithm="giac")
Output:
integrate(x*cot(b*x + a)^3, x)
Timed out. \[ \int x \cot ^3(a+b x) \, dx=\int x\,{\mathrm {cot}\left (a+b\,x\right )}^3 \,d x \] Input:
int(x*cot(a + b*x)^3,x)
Output:
int(x*cot(a + b*x)^3, x)
\[ \int x \cot ^3(a+b x) \, dx=\int \cot \left (b x +a \right )^{3} x d x \] Input:
int(x*cot(b*x+a)^3,x)
Output:
int(cot(a + b*x)**3*x,x)