Integrand size = 19, antiderivative size = 95 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a}{8 d (1-\cos (c+d x))^2}+\frac {3 a}{4 d (1-\cos (c+d x))}+\frac {a}{8 d (1+\cos (c+d x))}+\frac {11 a \log (1-\cos (c+d x))}{16 d}+\frac {5 a \log (1+\cos (c+d x))}{16 d} \] Output:
-1/8*a/d/(1-cos(d*x+c))^2+3/4*a/d/(1-cos(d*x+c))+1/8*a/d/(1+cos(d*x+c))+11 /16*a*ln(1-cos(d*x+c))/d+5/16*a*ln(1+cos(d*x+c))/d
Time = 0.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^2(c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \] Input:
Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]
Output:
(5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) + (a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^4)/(4*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d ) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) + (a*Log[Sin[c + d*x]])/d - (5*a*Sec [(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 25, 4367, 27, 99, 2009}
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.
\(\displaystyle \int \cot ^5(c+d x) (a \sec (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}{\cot \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle -\frac {a^6 \int \frac {\cos ^4(c+d x)}{a^5 (1-\cos (c+d x))^3 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \int \frac {\cos ^4(c+d x)}{(1-\cos (c+d x))^3 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {a \int \left (-\frac {5}{16 (\cos (c+d x)+1)}+\frac {1}{8 (\cos (c+d x)+1)^2}-\frac {11}{16 (\cos (c+d x)-1)}-\frac {3}{4 (\cos (c+d x)-1)^2}-\frac {1}{4 (\cos (c+d x)-1)^3}\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (-\frac {3}{4 (1-\cos (c+d x))}-\frac {1}{8 (\cos (c+d x)+1)}+\frac {1}{8 (1-\cos (c+d x))^2}-\frac {11}{16} \log (1-\cos (c+d x))-\frac {5}{16} \log (\cos (c+d x)+1)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]
Output:
-((a*(1/(8*(1 - Cos[c + d*x])^2) - 3/(4*(1 - Cos[c + d*x])) - 1/(8*(1 + Co s[c + d*x])) - (11*Log[1 - Cos[c + d*x]])/16 - (5*Log[1 + Cos[c + d*x]])/1 6))/d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(111\) |
default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(111\) |
risch | \(-i a x -\frac {2 i a c}{d}-\frac {a \left (5 \,{\mathrm e}^{5 i \left (d x +c \right )}+6 \,{\mathrm e}^{4 i \left (d x +c \right )}-14 \,{\mathrm e}^{3 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(139\) |
Input:
int(cot(d*x+c)^5*(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/4/sin(d*x+c)^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c)^5+1/8*c os(d*x+c)^3+3/8*cos(d*x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c)))+a*(-1/4*cot(d*x+ c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.58 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {10 \, a \cos \left (d x + c\right )^{2} + 6 \, a \cos \left (d x + c\right ) - 5 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 11 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/16*(10*a*cos(d*x + c)^2 + 6*a*cos(d*x + c) - 5*(a*cos(d*x + c)^3 - a*co s(d*x + c)^2 - a*cos(d*x + c) + a)*log(1/2*cos(d*x + c) + 1/2) - 11*(a*cos (d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(-1/2*cos(d*x + c) + 1/2) - 12*a)/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)
\[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=a \left (\int \cot ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**5*(a+a*sec(d*x+c)),x)
Output:
a*(Integral(cot(c + d*x)**5*sec(c + d*x), x) + Integral(cot(c + d*x)**5, x ))
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (5 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="maxima")
Output:
1/16*(5*a*log(cos(d*x + c) + 1) + 11*a*log(cos(d*x + c) - 1) - 2*(5*a*cos( d*x + c)^2 + 3*a*cos(d*x + c) - 6*a)/(cos(d*x + c)^3 - cos(d*x + c)^2 - co s(d*x + c) + 1))/d
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {1}{16} \, a {\left (\frac {5 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {11 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 6\right )}}{d {\left (\cos \left (d x + c\right ) + 1\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}\right )} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="giac")
Output:
1/16*a*(5*log(abs(cos(d*x + c) + 1))/d + 11*log(abs(cos(d*x + c) - 1))/d - 2*(5*cos(d*x + c)^2 + 3*cos(d*x + c) - 6)/(d*(cos(d*x + c) + 1)*(cos(d*x + c) - 1)^2))
Time = 12.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{4}-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}\right )}{8\,d}+\frac {11\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \] Input:
int(cot(c + d*x)^5*(a + a/cos(c + d*x)),x)
Output:
(a*tan(c/2 + (d*x)/2)^2)/(16*d) - (a*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (c ot(c/2 + (d*x)/2)^4*(a/4 - (5*a*tan(c/2 + (d*x)/2)^2)/2))/(8*d) + (11*a*lo g(tan(c/2 + (d*x)/2)))/(8*d)
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \left (-32 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+44 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}{32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+a*sec(d*x+c)),x)
Output:
(a*( - 32*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**4 + 44*log(tan((c + d*x)/2))*tan((c + d*x)/2)**4 + 2*tan((c + d*x)/2)**6 + 10*tan((c + d*x) /2)**2 - 1))/(32*tan((c + d*x)/2)**4*d)