Integrand size = 21, antiderivative size = 61 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3}{2 d (1-\cos (c+d x))^2}+\frac {2 a^3}{d (1-\cos (c+d x))}+\frac {a^3 \log (1-\cos (c+d x))}{d} \] Output:
-1/2*a^3/d/(1-cos(d*x+c))^2+2*a^3/d/(1-cos(d*x+c))+a^3*ln(1-cos(d*x+c))/d
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \left (-8 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )-16 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )}{64 d} \] Input:
Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]
Output:
-1/64*(a^3*(1 + Cos[c + d*x])^3*(-8*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^ 4 - 16*Log[Sin[(c + d*x)/2]])*Sec[(c + d*x)/2]^6)/d
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 49, 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.
\(\displaystyle \int \cot ^5(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle -\frac {a^6 \int \frac {\cos ^2(c+d x)}{a^3 (1-\cos (c+d x))^3}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^3 \int \frac {\cos ^2(c+d x)}{(1-\cos (c+d x))^3}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {a^3 \int \left (-\frac {2}{(\cos (c+d x)-1)^2}-\frac {1}{(\cos (c+d x)-1)^3}+\frac {1}{1-\cos (c+d x)}\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 \left (-\frac {2}{1-\cos (c+d x)}+\frac {1}{2 (1-\cos (c+d x))^2}-\log (1-\cos (c+d x))\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*(1/(2*(1 - Cos[c + d*x])^2) - 2/(1 - Cos[c + d*x]) - Log[1 - Cos[c + d*x]]))/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_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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]
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54
method | result | size |
risch | \(-i a^{3} x -\frac {2 i a^{3} c}{d}-\frac {2 a^{3} \left (2 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(94\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{4}}{4 \sin \left (d x +c \right )^{4}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(204\) |
default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{4}}{4 \sin \left (d x +c \right )^{4}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(204\) |
Input:
int(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-I*a^3*x-2*I/d*a^3*c-2*a^3/d/(exp(I*(d*x+c))-1)^4*(2*exp(3*I*(d*x+c))-3*ex p(2*I*(d*x+c))+2*exp(I*(d*x+c)))+2/d*a^3*ln(exp(I*(d*x+c))-1)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.34 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {4 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3} - 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/2*(4*a^3*cos(d*x + c) - 3*a^3 - 2*(a^3*cos(d*x + c)^2 - 2*a^3*cos(d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)
\[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \cot ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \cot ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**5*(a+a*sec(d*x+c))**3,x)
Output:
a**3*(Integral(3*cot(c + d*x)**5*sec(c + d*x), x) + Integral(3*cot(c + d*x )**5*sec(c + d*x)**2, x) + Integral(cot(c + d*x)**5*sec(c + d*x)**3, x) + Integral(cot(c + d*x)**5, x))
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {2 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {4 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3}}{\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}}{2 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(2*a^3*log(cos(d*x + c) - 1) - (4*a^3*cos(d*x + c) - 3*a^3)/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1))/d
Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1}{2} \, a^{3} {\left (\frac {2 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {4 \, \cos \left (d x + c\right ) - 3}{d {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}\right )} \] Input:
integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/2*a^3*(2*log(abs(cos(d*x + c) - 1))/d - (4*cos(d*x + c) - 3)/(d*(cos(d*x + c) - 1)^2))
Time = 12.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {2\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {a^3}{8}}{d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cot(c + d*x)^5*(a + a/cos(c + d*x))^3,x)
Output:
(2*a^3*log(tan(c/2 + (d*x)/2)))/d + ((3*a^3*tan(c/2 + (d*x)/2)^2)/4 - a^3/ 8)/(d*tan(c/2 + (d*x)/2)^4) - (a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.38 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+16 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 8*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**4 + 16*log(tan( (c + d*x)/2))*tan((c + d*x)/2)**4 + 6*tan((c + d*x)/2)**2 - 1))/(8*tan((c + d*x)/2)**4*d)