\(\int (\frac {1}{2}-3 \cot (x)) (3-2 \cot (x))^3 \, dx\) [365]

Optimal result

Integrand size = 17, antiderivative size = 33 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=-\frac {285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \log (\sin (x)) \]

[Out]

-285/2*x+5*(3-2*cot(x))^2+(3-2*cot(x))^3-42*cot(x)+4*ln(sin(x))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3609, 3606, 3556} \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=-\frac {285 x}{2}+(3-2 \cot (x))^3+5 (3-2 \cot (x))^2-42 \cot (x)+4 \log (\sin (x)) \]

[In]

Int[(1/2 - 3*Cot[x])*(3 - 2*Cot[x])^3,x]

[Out]

(-285*x)/2 + 5*(3 - 2*Cot[x])^2 + (3 - 2*Cot[x])^3 - 42*Cot[x] + 4*Log[Sin[x]]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = (3-2 \cot (x))^3+\int \left (-\frac {9}{2}-10 \cot (x)\right ) (3-2 \cot (x))^2 \, dx \\ & = 5 (3-2 \cot (x))^2+(3-2 \cot (x))^3+\int \left (-\frac {67}{2}-21 \cot (x)\right ) (3-2 \cot (x)) \, dx \\ & = -\frac {285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \int \cot (x) \, dx \\ & = -\frac {285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \log (\sin (x)) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=\frac {27 x}{2}+56 \cot ^2(x)-8 \cot ^3(x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(x)\right )-180 \cot (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(x)\right )+4 \log (\cos (x))+4 \log (\tan (x)) \]

[In]

Integrate[(1/2 - 3*Cot[x])*(3 - 2*Cot[x])^3,x]

[Out]

(27*x)/2 + 56*Cot[x]^2 - 8*Cot[x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[x]^2] - 180*Cot[x]*Hypergeometric2F1
[-1/2, 1, 1/2, -Tan[x]^2] + 4*Log[Cos[x]] + 4*Log[Tan[x]]

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(4 \ln \left (\tan \left (x \right )\right )-2 \ln \left (\sec ^{2}\left (x \right )\right )-\frac {285 x}{2}-8 \left (\cot ^{3}\left (x \right )\right )-156 \cot \left (x \right )+56 \left (\cot ^{2}\left (x \right )\right )\)	\(33\)
derivativedivides	\(-8 \left (\cot ^{3}\left (x \right )\right )+56 \left (\cot ^{2}\left (x \right )\right )-156 \cot \left (x \right )-2 \ln \left (\cot ^{2}\left (x \right )+1\right )+\frac {285 \pi }{4}-\frac {285 \,\operatorname {arccot}\left (\cot \left (x \right )\right )}{2}\)	\(35\)
default	\(-8 \left (\cot ^{3}\left (x \right )\right )+56 \left (\cot ^{2}\left (x \right )\right )-156 \cot \left (x \right )-2 \ln \left (\cot ^{2}\left (x \right )+1\right )+\frac {285 \pi }{4}-\frac {285 \,\operatorname {arccot}\left (\cot \left (x \right )\right )}{2}\)	\(35\)
norman	\(\frac {-8-156 \left (\tan ^{2}\left (x \right )\right )-\frac {285 x \left (\tan ^{3}\left (x \right )\right )}{2}+56 \tan \left (x \right )}{\tan \left (x \right )^{3}}+4 \ln \left (\tan \left (x \right )\right )-2 \ln \left (1+\tan ^{2}\left (x \right )\right )\)	\(40\)
parts	\(\frac {27 x}{2}-156 \cot \left (x \right )+78 \pi -156 \,\operatorname {arccot}\left (\cot \left (x \right )\right )+56 \left (\cot ^{2}\left (x \right )\right )-56 \ln \left (\cot ^{2}\left (x \right )+1\right )-8 \left (\cot ^{3}\left (x \right )\right )-108 \ln \left (\sin \left (x \right )\right )\)	\(43\)
risch	\(-\frac {285 x}{2}-4 i x +\frac {\left (-\frac {224}{1873}-\frac {264 i}{1873}\right ) \left (1873 \,{\mathrm e}^{4 i x}-1260 i {\mathrm e}^{2 i x}-3358 \,{\mathrm e}^{2 i x}+1221+1036 i\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{3}}+4 \ln \left ({\mathrm e}^{2 i x}-1\right )\)	\(58\)

[In]

int((1/2-3*cot(x))*(3-2*cot(x))^3,x,method=_RETURNVERBOSE)

[Out]

4*ln(tan(x))-2*ln(sec(x)^2)-285/2*x-8*cot(x)^3-156*cot(x)+56*cot(x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.15 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=\frac {4 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (2 \, x\right ) - 296 \, \cos \left (2 \, x\right )^{2} - {\left (285 \, x \cos \left (2 \, x\right ) - 285 \, x + 224\right )} \sin \left (2 \, x\right ) + 32 \, \cos \left (2 \, x\right ) + 328}{2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]

[In]

integrate((1/2-3*cot(x))*(3-2*cot(x))^3,x, algorithm="fricas")

[Out]

1/2*(4*(cos(2*x) - 1)*log(-1/2*cos(2*x) + 1/2)*sin(2*x) - 296*cos(2*x)^2 - (285*x*cos(2*x) - 285*x + 224)*sin(
2*x) + 32*cos(2*x) + 328)/((cos(2*x) - 1)*sin(2*x))

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=- \frac {285 x}{2} - 2 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} + 4 \log {\left (\tan {\left (x \right )} \right )} - \frac {156}{\tan {\left (x \right )}} + \frac {56}{\tan ^{2}{\left (x \right )}} - \frac {8}{\tan ^{3}{\left (x \right )}} \]

[In]

integrate((1/2-3*cot(x))*(3-2*cot(x))**3,x)

[Out]

-285*x/2 - 2*log(tan(x)**2 + 1) + 4*log(tan(x)) - 156/tan(x) + 56/tan(x)**2 - 8/tan(x)**3

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=-\frac {285}{2} \, x - \frac {4 \, {\left (39 \, \tan \left (x\right )^{2} - 14 \, \tan \left (x\right ) + 2\right )}}{\tan \left (x\right )^{3}} - 2 \, \log \left (\tan \left (x\right )^{2} + 1\right ) + 4 \, \log \left (\tan \left (x\right )\right ) \]

[In]

integrate((1/2-3*cot(x))*(3-2*cot(x))^3,x, algorithm="maxima")

[Out]

-285/2*x - 4*(39*tan(x)^2 - 14*tan(x) + 2)/tan(x)^3 - 2*log(tan(x)^2 + 1) + 4*log(tan(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (33) = 66\).

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=\tan \left (\frac {1}{2} \, x\right )^{3} + 14 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {285}{2} \, x - \frac {22 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 225 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 42 \, \tan \left (\frac {1}{2} \, x\right ) + 3}{3 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - 4 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 4 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) + 75 \, \tan \left (\frac {1}{2} \, x\right ) \]

[In]

integrate((1/2-3*cot(x))*(3-2*cot(x))^3,x, algorithm="giac")

[Out]

tan(1/2*x)^3 + 14*tan(1/2*x)^2 - 285/2*x - 1/3*(22*tan(1/2*x)^3 + 225*tan(1/2*x)^2 - 42*tan(1/2*x) + 3)/tan(1/
2*x)^3 - 4*log(tan(1/2*x)^2 + 1) + 4*log(abs(tan(1/2*x))) + 75*tan(1/2*x)

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx=x\,\left (-\frac {285}{2}-4{}\mathrm {i}\right )+4\,\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}-1\right )+\frac {64{}\mathrm {i}}{3\,{\mathrm {e}}^{x\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}-1}+\frac {-224+96{}\mathrm {i}}{1+{\mathrm {e}}^{x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{x\,2{}\mathrm {i}}}+\frac {-224-264{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1} \]

[In]

int((2*cot(x) - 3)^3*(3*cot(x) - 1/2),x)

[Out]

4*log(exp(x*2i) - 1) - x*(285/2 + 4i) + 64i/(3*exp(x*2i) - 3*exp(x*4i) + exp(x*6i) - 1) - (224 - 96i)/(exp(x*4
i) - 2*exp(x*2i) + 1) - (224 + 264i)/(exp(x*2i) - 1)