Integrand size = 21, antiderivative size = 107 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3}{6 d (1-\cos (c+d x))^3}+\frac {7 a^3}{8 d (1-\cos (c+d x))^2}-\frac {17 a^3}{8 d (1-\cos (c+d x))}-\frac {15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {a^3 \log (1+\cos (c+d x))}{16 d} \] Output:
-1/6*a^3/d/(1-cos(d*x+c))^3+7/8*a^3/d/(1-cos(d*x+c))^2-17/8*a^3/d/(1-cos(d *x+c))-15/16*a^3*ln(1-cos(d*x+c))/d-1/16*a^3*ln(1+cos(d*x+c))/d
Time = 0.47 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \left (102 \csc ^2\left (\frac {1}{2} (c+d x)\right )-21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+2 \csc ^6\left (\frac {1}{2} (c+d x)\right )+12 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )}{768 d} \] Input:
Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]
Output:
-1/768*(a^3*(1 + Cos[c + d*x])^3*(102*Csc[(c + d*x)/2]^2 - 21*Csc[(c + d*x )/2]^4 + 2*Csc[(c + d*x)/2]^6 + 12*(Log[Cos[(c + d*x)/2]] + 15*Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^6)/d
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79, 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.
| \(\displaystyle \int \cot ^7(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^7}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {a^8 \int \frac {\cos ^4(c+d x)}{a^5 (1-\cos (c+d x))^4 (\cos (c+d x)+1)}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos ^4(c+d x)}{(1-\cos (c+d x))^4 (\cos (c+d x)+1)}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^3 \int \left (\frac {1}{16 (\cos (c+d x)+1)}+\frac {15}{16 (\cos (c+d x)-1)}+\frac {17}{8 (\cos (c+d x)-1)^2}+\frac {7}{4 (\cos (c+d x)-1)^3}+\frac {1}{2 (\cos (c+d x)-1)^4}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \left (\frac {17}{8 (1-\cos (c+d x))}-\frac {7}{8 (1-\cos (c+d x))^2}+\frac {1}{6 (1-\cos (c+d x))^3}+\frac {15}{16} \log (1-\cos (c+d x))+\frac {1}{16} \log (\cos (c+d x)+1)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*(1/(6*(1 - Cos[c + d*x])^3) - 7/(8*(1 - Cos[c + d*x])^2) + 17/(8*(1 - Cos[c + d*x])) + (15*Log[1 - Cos[c + d*x]])/16 + 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 = 2.71 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(i a^{3} x +\frac {2 i a^{3} c}{d}+\frac {a^{3} \left (51 \,{\mathrm e}^{5 i \left (d x +c \right )}-162 \,{\mathrm e}^{4 i \left (d x +c \right )}+238 \,{\mathrm e}^{3 i \left (d x +c \right )}-162 \,{\mathrm e}^{2 i \left (d x +c \right )}+51 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(136\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{6}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(272\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{6}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(272\) |
Input:
int(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
I*a^3*x+2*I/d*a^3*c+1/12*a^3/d/(exp(I*(d*x+c))-1)^6*(51*exp(5*I*(d*x+c))-1 62*exp(4*I*(d*x+c))+238*exp(3*I*(d*x+c))-162*exp(2*I*(d*x+c))+51*exp(I*(d* x+c)))-1/8/d*a^3*ln(exp(I*(d*x+c))+1)-15/8/d*a^3*ln(exp(I*(d*x+c))-1)
Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.66 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {102 \, a^{3} \cos \left (d x + c\right )^{2} - 162 \, a^{3} \cos \left (d x + c\right ) + 68 \, a^{3} - 3 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )}} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/48*(102*a^3*cos(d*x + c)^2 - 162*a^3*cos(d*x + c) + 68*a^3 - 3*(a^3*cos( d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) - a^3)*log(1/2*cos( d*x + c) + 1/2) - 45*(a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + 3*a^3*co s(d*x + c) - a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^3 - 3*d*co s(d*x + c)^2 + 3*d*cos(d*x + c) - d)
Timed out. \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 45 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (51 \, a^{3} \cos \left (d x + c\right )^{2} - 81 \, a^{3} \cos \left (d x + c\right ) + 34 \, a^{3}\right )}}{\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 1}}{48 \, d} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/48*(3*a^3*log(cos(d*x + c) + 1) + 45*a^3*log(cos(d*x + c) - 1) - 2*(51* a^3*cos(d*x + c)^2 - 81*a^3*cos(d*x + c) + 34*a^3)/(cos(d*x + c)^3 - 3*cos (d*x + c)^2 + 3*cos(d*x + c) - 1))/d
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.66 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {1}{48} \, a^{3} {\left (\frac {3 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {45 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, {\left (51 \, \cos \left (d x + c\right )^{2} - 81 \, \cos \left (d x + c\right ) + 34\right )}}{d {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}\right )} \] Input:
integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
-1/48*a^3*(3*log(abs(cos(d*x + c) + 1))/d + 45*log(abs(cos(d*x + c) - 1))/ d - 2*(51*cos(d*x + c)^2 - 81*cos(d*x + c) + 34)/(d*(cos(d*x + c) - 1)^3))
Time = 12.67 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {a^3}{6}}{8\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}-\frac {15\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \] Input:
int(cot(c + d*x)^7*(a + a/cos(c + d*x))^3,x)
Output:
(a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d - ((11*a^3*tan(c/2 + (d*x)/2)^4)/2 - (5*a^3*tan(c/2 + (d*x)/2)^2)/4 + a^3/6)/(8*d*tan(c/2 + (d*x)/2)^6) - (15* a^3*log(tan(c/2 + (d*x)/2)))/(8*d)
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91 \[ \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-180 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-66 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2\right )}{96 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} d} \] Input:
int(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*(96*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**6 - 180*log(tan(( c + d*x)/2))*tan((c + d*x)/2)**6 - 66*tan((c + d*x)/2)**4 + 15*tan((c + d* x)/2)**2 - 2))/(96*tan((c + d*x)/2)**6*d)