Integrand size = 21, antiderivative size = 127 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2}{12 d (1-\cos (c+d x))^3}+\frac {a^2}{2 d (1-\cos (c+d x))^2}-\frac {23 a^2}{16 d (1-\cos (c+d x))}-\frac {a^2}{16 d (1+\cos (c+d x))}-\frac {13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac {3 a^2 \log (1+\cos (c+d x))}{16 d} \] Output:
-1/12*a^2/d/(1-cos(d*x+c))^3+1/2*a^2/d/(1-cos(d*x+c))^2-23/16*a^2/d/(1-cos (d*x+c))-1/16*a^2/d/(1+cos(d*x+c))-13/16*a^2*ln(1-cos(d*x+c))/d-3/16*a^2*l n(1+cos(d*x+c))/d
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (69 \csc ^2\left (\frac {1}{2} (c+d x)\right )-12 \csc ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \left (12 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+52 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )}{384 d} \] Input:
Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]
Output:
-1/384*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(69*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6 + 3*(12*Log[Cos[(c + d*x)/2] ] + 52*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2)))/d
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78, 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, 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.
| \(\displaystyle \int \cot ^7(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^2}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^7}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {a^8 \int \frac {\cos ^5(c+d x)}{a^6 (1-\cos (c+d x))^4 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \int \frac {\cos ^5(c+d x)}{(1-\cos (c+d x))^4 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^2 \int \left (\frac {3}{16 (\cos (c+d x)+1)}-\frac {1}{16 (\cos (c+d x)+1)^2}+\frac {13}{16 (\cos (c+d x)-1)}+\frac {23}{16 (\cos (c+d x)-1)^2}+\frac {1}{(\cos (c+d x)-1)^3}+\frac {1}{4 (\cos (c+d x)-1)^4}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \left (\frac {23}{16 (1-\cos (c+d x))}+\frac {1}{16 (\cos (c+d x)+1)}-\frac {1}{2 (1-\cos (c+d x))^2}+\frac {1}{12 (1-\cos (c+d x))^3}+\frac {13}{16} \log (1-\cos (c+d x))+\frac {3}{16} \log (\cos (c+d x)+1)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]
Output:
-((a^2*(1/(12*(1 - Cos[c + d*x])^3) - 1/(2*(1 - Cos[c + d*x])^2) + 23/(16* (1 - Cos[c + d*x])) + 1/(16*(1 + Cos[c + d*x])) + (13*Log[1 - Cos[c + d*x] ])/16 + (3*Log[1 + Cos[c + d*x]])/16))/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]
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(i a^{2} x +\frac {2 i a^{2} c}{d}+\frac {a^{2} \left (33 \,{\mathrm e}^{7 i \left (d x +c \right )}-36 \,{\mathrm e}^{6 i \left (d x +c \right )}-49 \,{\mathrm e}^{5 i \left (d x +c \right )}+136 \,{\mathrm e}^{4 i \left (d x +c \right )}-49 \,{\mathrm e}^{3 i \left (d x +c \right )}-36 \,{\mathrm e}^{2 i \left (d x +c \right )}+33 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}-\frac {13 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(171\) |
| derivativedivides | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{6}}{6 \sin \left (d x +c \right )^{6}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(177\) |
| default | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{6}}{6 \sin \left (d x +c \right )^{6}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(177\) |
Input:
int(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
I*a^2*x+2*I/d*a^2*c+1/12*a^2/d/(exp(I*(d*x+c))-1)^6/(exp(I*(d*x+c))+1)^2*( 33*exp(7*I*(d*x+c))-36*exp(6*I*(d*x+c))-49*exp(5*I*(d*x+c))+136*exp(4*I*(d *x+c))-49*exp(3*I*(d*x+c))-36*exp(2*I*(d*x+c))+33*exp(I*(d*x+c)))-13/8/d*a ^2*ln(exp(I*(d*x+c))-1)-3/8/d*a^2*ln(exp(I*(d*x+c))+1)
Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.50 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {66 \, a^{2} \cos \left (d x + c\right )^{3} - 36 \, a^{2} \cos \left (d x + c\right )^{2} - 74 \, a^{2} \cos \left (d x + c\right ) + 52 \, a^{2} - 9 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 39 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) - d\right )}} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Output:
1/48*(66*a^2*cos(d*x + c)^3 - 36*a^2*cos(d*x + c)^2 - 74*a^2*cos(d*x + c) + 52*a^2 - 9*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(1/2*cos(d*x + c) + 1/2) - 39*(a^2*cos(d*x + c)^4 - 2*a^2*cos (d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*c os(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c) - d)
\[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=a^{2} \left (\int 2 \cot ^{7}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**7*(a+a*sec(d*x+c))**2,x)
Output:
a**2*(Integral(2*cot(c + d*x)**7*sec(c + d*x), x) + Integral(cot(c + d*x)* *7*sec(c + d*x)**2, x) + Integral(cot(c + d*x)**7, x))
Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {9 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) + 39 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (33 \, a^{2} \cos \left (d x + c\right )^{3} - 18 \, a^{2} \cos \left (d x + c\right )^{2} - 37 \, a^{2} \cos \left (d x + c\right ) + 26 \, a^{2}\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right ) - 1}}{48 \, d} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm="maxima")
Output:
-1/48*(9*a^2*log(cos(d*x + c) + 1) + 39*a^2*log(cos(d*x + c) - 1) - 2*(33* a^2*cos(d*x + c)^3 - 18*a^2*cos(d*x + c)^2 - 37*a^2*cos(d*x + c) + 26*a^2) /(cos(d*x + c)^4 - 2*cos(d*x + c)^3 + 2*cos(d*x + c) - 1))/d
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.72 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {1}{48} \, a^{2} {\left (\frac {9 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {39 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{3} - 18 \, \cos \left (d x + c\right )^{2} - 37 \, \cos \left (d x + c\right ) + 26\right )}}{d {\left (\cos \left (d x + c\right ) + 1\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}\right )} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm="giac")
Output:
-1/48*a^2*(9*log(abs(cos(d*x + c) + 1))/d + 39*log(abs(cos(d*x + c) - 1))/ d - 2*(33*cos(d*x + c)^3 - 18*cos(d*x + c)^2 - 37*cos(d*x + c) + 26)/(d*(c os(d*x + c) + 1)*(cos(d*x + c) - 1)^3))
Time = 12.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {13\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {a^2}{6}\right )}{16\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32\,d} \] Input:
int(cot(c + d*x)^7*(a + a/cos(c + d*x))^2,x)
Output:
(a^2*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (13*a^2*log(tan(c/2 + (d*x)/2)))/( 8*d) - (cot(c/2 + (d*x)/2)^6*(8*a^2*tan(c/2 + (d*x)/2)^4 - (3*a^2*tan(c/2 + (d*x)/2)^2)/2 + a^2/6))/(16*d) - (a^2*tan(c/2 + (d*x)/2)^2)/(32*d)
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^{2} \left (96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-156 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}{96 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} d} \] Input:
int(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x)
Output:
(a**2*(96*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**6 - 156*log(tan(( c + d*x)/2))*tan((c + d*x)/2)**6 - 3*tan((c + d*x)/2)**8 - 48*tan((c + d*x )/2)**4 + 9*tan((c + d*x)/2)**2 - 1))/(96*tan((c + d*x)/2)**6*d)