Integrand size = 13, antiderivative size = 46 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {3 \text {arctanh}(\cos (x))}{8 a}-\frac {\cot ^4(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a}+\frac {\cot ^3(x) \csc (x)}{4 a} \] Output:
3/8*arctanh(cos(x))/a-1/4*cot(x)^4/a-3/8*cot(x)*csc(x)/a+1/4*cot(x)^3*csc( x)/a
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=-\frac {-8+2 \cot ^2\left (\frac {x}{2}\right )-12 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+\sec ^2\left (\frac {x}{2}\right )}{16 a (1+\cos (x))} \] Input:
Integrate[Cot[x]^3/(a + a*Cos[x]),x]
Output:
-1/16*(-8 + 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(a*(1 + Cos[x]))
Time = 0.45 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 3185, 25, 3042, 25, 3087, 15, 3091, 3042, 3091, 3042, 4257}
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.
\(\displaystyle \int \frac {\cot ^3(x)}{a \cos (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (x-\frac {\pi }{2}\right )^3}{a-a \sin \left (x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (x-\frac {\pi }{2}\right )^3}{a-a \sin \left (x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle -\frac {\int \cot ^4(x) \csc (x)dx}{a}-\frac {\int -\cot ^3(x) \csc ^2(x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot ^3(x) \csc ^2(x)dx}{a}-\frac {\int \cot ^4(x) \csc (x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^3dx}{a}-\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^3dx}{a}-\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {\int -\cot ^3(x)d(-\cot (x))}{a}-\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}-\frac {\cot ^4(x)}{4 a}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {3}{4} \int \cot ^2(x) \csc (x)dx-\frac {1}{4} \cot ^3(x) \csc (x)}{a}-\frac {\cot ^4(x)}{4 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {3}{4} \int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^2dx-\frac {1}{4} \cot ^3(x) \csc (x)}{a}-\frac {\cot ^4(x)}{4 a}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {3}{4} \left (-\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot ^3(x) \csc (x)}{a}-\frac {\cot ^4(x)}{4 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {3}{4} \left (-\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot ^3(x) \csc (x)}{a}-\frac {\cot ^4(x)}{4 a}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {-\frac {3}{4} \left (\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot ^3(x) \csc (x)}{a}-\frac {\cot ^4(x)}{4 a}\) |
Input:
Int[Cot[x]^3/(a + a*Cos[x]),x]
Output:
-1/4*Cot[x]^4/a - (-1/4*(Cot[x]^3*Csc[x]) - (3*(ArcTanh[Cos[x]]/2 - (Cot[x ]*Csc[x])/2))/4)/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {-\frac {1}{8 \left (\cos \left (x \right )+1\right )^{2}}+\frac {1}{2 \cos \left (x \right )+2}+\frac {3 \ln \left (\cos \left (x \right )+1\right )}{16}+\frac {1}{-8+8 \cos \left (x \right )}-\frac {3 \ln \left (-1+\cos \left (x \right )\right )}{16}}{a}\) | \(44\) |
risch | \(\frac {5 \,{\mathrm e}^{5 i x}+2 \,{\mathrm e}^{4 i x}+2 \,{\mathrm e}^{3 i x}+2 \,{\mathrm e}^{2 i x}+5 \,{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{i x}+1\right )^{4} a \left ({\mathrm e}^{i x}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{8 a}\) | \(87\) |
Input:
int(cot(x)^3/(a+a*cos(x)),x,method=_RETURNVERBOSE)
Output:
1/a*(-1/8/(cos(x)+1)^2+1/2/(cos(x)+1)+3/16*ln(cos(x)+1)+1/8/(-1+cos(x))-3/ 16*ln(-1+cos(x)))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (38) = 76\).
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {10 \, \cos \left (x\right )^{2} + 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (x\right ) - 4}{16 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} \] Input:
integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="fricas")
Output:
1/16*(10*cos(x)^2 + 3*(cos(x)^3 + cos(x)^2 - cos(x) - 1)*log(1/2*cos(x) + 1/2) - 3*(cos(x)^3 + cos(x)^2 - cos(x) - 1)*log(-1/2*cos(x) + 1/2) + 2*cos (x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)
\[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {\int \frac {\cot ^{3}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(x)**3/(a+a*cos(x)),x)
Output:
Integral(cot(x)**3/(cos(x) + 1), x)/a
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} + \frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac {3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \] Input:
integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="maxima")
Output:
1/8*(5*cos(x)^2 + cos(x) - 2)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a) + 3 /16*log(cos(x) + 1)/a - 3/16*log(cos(x) - 1)/a
Time = 0.37 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac {3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac {5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \, a {\left (\cos \left (x\right ) + 1\right )}^{2} {\left (\cos \left (x\right ) - 1\right )}} \] Input:
integrate(cot(x)^3/(a+a*cos(x)),x, algorithm="giac")
Output:
3/16*log(cos(x) + 1)/a - 3/16*log(-cos(x) + 1)/a + 1/8*(5*cos(x)^2 + cos(x ) - 2)/(a*(cos(x) + 1)^2*(cos(x) - 1))
Time = 40.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^6-6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+2}{32\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \] Input:
int(cot(x)^3/(a + a*cos(x)),x)
Output:
-(tan(x/2)^6 - 6*tan(x/2)^4 + 12*tan(x/2)^2*log(tan(x/2)) + 2)/(32*a*tan(x /2)^2)
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {-12 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{2}-\tan \left (\frac {x}{2}\right )^{6}+6 \tan \left (\frac {x}{2}\right )^{4}-2}{32 \tan \left (\frac {x}{2}\right )^{2} a} \] Input:
int(cot(x)^3/(a+a*cos(x)),x)
Output:
( - 12*log(tan(x/2))*tan(x/2)**2 - tan(x/2)**6 + 6*tan(x/2)**4 - 2)/(32*ta n(x/2)**2*a)