Integrand size = 13, antiderivative size = 40 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=-\frac {\cot ^5(x)}{5 a}+\frac {\csc (x)}{a}-\frac {2 \csc ^3(x)}{3 a}+\frac {\csc ^5(x)}{5 a} \] Output:
-1/5*cot(x)^5/a+csc(x)/a-2/3*csc(x)^3/a+1/5*csc(x)^5/a
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=-\frac {(-25+8 \cos (x)+36 \cos (2 x)+24 \cos (3 x)-3 \cos (4 x)) \csc ^3(x)}{120 a (1+\cos (x))} \] Input:
Integrate[Cot[x]^4/(a + a*Cos[x]),x]
Output:
-1/120*((-25 + 8*Cos[x] + 36*Cos[2*x] + 24*Cos[3*x] - 3*Cos[4*x])*Csc[x]^3 )/(a*(1 + Cos[x]))
Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 3185, 25, 3042, 25, 3086, 210, 2009, 3087, 15}
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.
\(\displaystyle \int \frac {\cot ^4(x)}{a \cos (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (x-\frac {\pi }{2}\right )^4}{a-a \sin \left (x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle \frac {\int -\cot ^5(x) \csc (x)dx}{a}+\frac {\int \cot ^4(x) \csc ^2(x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot ^4(x) \csc ^2(x)dx}{a}-\frac {\int \cot ^5(x) \csc (x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}-\frac {\int -\sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^5dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}+\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \tan \left (x-\frac {\pi }{2}\right )^5dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {\int \left (\csc ^2(x)-1\right )^2d\csc (x)}{a}+\frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle \frac {\int \left (\csc ^4(x)-2 \csc ^2(x)+1\right )d\csc (x)}{a}+\frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\int \sec \left (x-\frac {\pi }{2}\right )^2 \tan \left (x-\frac {\pi }{2}\right )^4dx}{a}+\frac {\frac {\csc ^5(x)}{5}-\frac {2 \csc ^3(x)}{3}+\csc (x)}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\int \cot ^4(x)d(-\cot (x))}{a}+\frac {\frac {\csc ^5(x)}{5}-\frac {2 \csc ^3(x)}{3}+\csc (x)}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\frac {\csc ^5(x)}{5}-\frac {2 \csc ^3(x)}{3}+\csc (x)}{a}-\frac {\cot ^5(x)}{5 a}\) |
Input:
Int[Cot[x]^4/(a + a*Cos[x]),x]
Output:
-1/5*Cot[x]^5/a + (Csc[x] - (2*Csc[x]^3)/3 + Csc[x]^5/5)/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 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[((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]
Time = 0.47 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {\frac {\tan \left (\frac {x}{2}\right )^{5}}{5}-\frac {4 \tan \left (\frac {x}{2}\right )^{3}}{3}+6 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {x}{2}\right )}}{16 a}\) | \(45\) |
risch | \(\frac {2 i \left (15 \,{\mathrm e}^{7 i x}+15 \,{\mathrm e}^{6 i x}-5 \,{\mathrm e}^{5 i x}-25 \,{\mathrm e}^{4 i x}+13 \,{\mathrm e}^{3 i x}+21 \,{\mathrm e}^{2 i x}+9 \,{\mathrm e}^{i x}-3\right )}{15 \left ({\mathrm e}^{i x}+1\right )^{5} a \left ({\mathrm e}^{i x}-1\right )^{3}}\) | \(76\) |
Input:
int(cot(x)^4/(a+a*cos(x)),x,method=_RETURNVERBOSE)
Output:
1/16/a*(1/5*tan(1/2*x)^5-4/3*tan(1/2*x)^3+6*tan(1/2*x)-1/3/tan(1/2*x)^3+4/ tan(1/2*x))
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=-\frac {3 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{3} - 12 \, \cos \left (x\right )^{2} + 8 \, \cos \left (x\right ) + 8}{15 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )} \sin \left (x\right )} \] Input:
integrate(cot(x)^4/(a+a*cos(x)),x, algorithm="fricas")
Output:
-1/15*(3*cos(x)^4 - 12*cos(x)^3 - 12*cos(x)^2 + 8*cos(x) + 8)/((a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)*sin(x))
\[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=\frac {\int \frac {\cot ^{4}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(x)**4/(a+a*cos(x)),x)
Output:
Integral(cot(x)**4/(cos(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (34) = 68\).
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.75 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=\frac {\frac {90 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {20 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{240 \, a} + \frac {{\left (\frac {12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{48 \, a \sin \left (x\right )^{3}} \] Input:
integrate(cot(x)^4/(a+a*cos(x)),x, algorithm="maxima")
Output:
1/240*(90*sin(x)/(cos(x) + 1) - 20*sin(x)^3/(cos(x) + 1)^3 + 3*sin(x)^5/(c os(x) + 1)^5)/a + 1/48*(12*sin(x)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)^3/(a* sin(x)^3)
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=\frac {12 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{48 \, a \tan \left (\frac {1}{2} \, x\right )^{3}} + \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{5} - 20 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 90 \, a^{4} \tan \left (\frac {1}{2} \, x\right )}{240 \, a^{5}} \] Input:
integrate(cot(x)^4/(a+a*cos(x)),x, algorithm="giac")
Output:
1/48*(12*tan(1/2*x)^2 - 1)/(a*tan(1/2*x)^3) + 1/240*(3*a^4*tan(1/2*x)^5 - 20*a^4*tan(1/2*x)^3 + 90*a^4*tan(1/2*x))/a^5
Time = 41.74 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+60\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-5}{240\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \] Input:
int(cot(x)^4/(a + a*cos(x)),x)
Output:
(60*tan(x/2)^2 + 90*tan(x/2)^4 - 20*tan(x/2)^6 + 3*tan(x/2)^8 - 5)/(240*a* tan(x/2)^3)
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^4(x)}{a+a \cos (x)} \, dx=\frac {12 \cos \left (x \right ) \sin \left (x \right )^{2}-4 \cos \left (x \right )+3 \sin \left (x \right )^{4}+6 \sin \left (x \right )^{2}-1}{15 \sin \left (x \right )^{3} a \left (\cos \left (x \right )+1\right )} \] Input:
int(cot(x)^4/(a+a*cos(x)),x)
Output:
(12*cos(x)*sin(x)**2 - 4*cos(x) + 3*sin(x)**4 + 6*sin(x)**2 - 1)/(15*sin(x )**3*a*(cos(x) + 1))