Integrand size = 19, antiderivative size = 130 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac {(16 a-5 b) \log (1+\cos (c+d x))}{32 d}-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \] Output:
-1/32*(16*a+5*b)*ln(1-cos(d*x+c))/d-1/32*(16*a-5*b)*ln(1+cos(d*x+c))/d-1/6 *cot(d*x+c)^6*(a+b*sec(d*x+c))/d+1/24*cot(d*x+c)^4*(6*a+5*b*sec(d*x+c))/d- 1/16*cot(d*x+c)^2*(8*a+5*b*sec(d*x+c))/d
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.69 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {11 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {b \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {b \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {5 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \log (\sin (c+d x))}{d}+\frac {11 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {b \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \] Input:
Integrate[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]
Output:
(-11*b*Csc[(c + d*x)/2]^2)/(64*d) + (b*Csc[(c + d*x)/2]^4)/(32*d) - (b*Csc [(c + d*x)/2]^6)/(384*d) - (3*a*Csc[c + d*x]^2)/(2*d) + (3*a*Csc[c + d*x]^ 4)/(4*d) - (a*Csc[c + d*x]^6)/(6*d) + (5*b*Log[Cos[(c + d*x)/2]])/(16*d) - (5*b*Log[Sin[(c + d*x)/2]])/(16*d) - (a*Log[Sin[c + d*x]])/d + (11*b*Sec[ (c + d*x)/2]^2)/(64*d) - (b*Sec[(c + d*x)/2]^4)/(32*d) + (b*Sec[(c + d*x)/ 2]^6)/(384*d)
Time = 0.74 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 4370, 3042, 25, 4370, 27, 3042, 25, 4370, 3042, 25, 4371, 3042, 25, 3147, 452, 219, 240}
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.
| \(\displaystyle \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^7}dx\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {1}{6} \int \cot ^5(c+d x) (6 a+5 b \sec (c+d x))dx-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{6} \int -\frac {6 a+5 b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^5}dx-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \int \frac {6 a+5 b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}dx-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int 3 \cot ^3(c+d x) (8 a+5 b \sec (c+d x))dx+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int \cot ^3(c+d x) (8 a+5 b \sec (c+d x))dx+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int -\frac {8 a+5 b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \int \frac {8 a+5 b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \cot (c+d x) (16 a+5 b \sec (c+d x))dx+\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int -\frac {16 a+5 b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {1}{2} \int \frac {16 a+5 b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 4371 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {1}{2} \int \sec \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (5 b+16 a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )dx\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {1}{2} \int -\frac {5 b-16 a \sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )}dx\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \frac {5 b-16 a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx+\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {8 a \int \frac {5 b+16 a \cos (c+d x)}{256 a^2-256 a^2 \cos ^2(c+d x)}d(16 a \cos (c+d x))}{d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {8 a \left (5 b \int \frac {1}{256 a^2-256 a^2 \cos ^2(c+d x)}d(16 a \cos (c+d x))+\int \frac {16 a \cos (c+d x)}{256 a^2-256 a^2 \cos ^2(c+d x)}d(16 a \cos (c+d x))\right )}{d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {8 a \left (\int \frac {16 a \cos (c+d x)}{256 a^2-256 a^2 \cos ^2(c+d x)}d(16 a \cos (c+d x))+\frac {5 b \text {arctanh}(\cos (c+d x))}{16 a}\right )}{d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {1}{6} \left (\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{4 d}-\frac {3}{4} \left (\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{2 d}-\frac {8 a \left (\frac {5 b \text {arctanh}(\cos (c+d x))}{16 a}-\frac {1}{2} \log \left (256 a^2-256 a^2 \cos ^2(c+d x)\right )\right )}{d}\right )\right )-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}\) |
Input:
Int[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]
Output:
-1/6*(Cot[c + d*x]^6*(a + b*Sec[c + d*x]))/d + ((Cot[c + d*x]^4*(6*a + 5*b *Sec[c + d*x]))/(4*d) - (3*((-8*a*((5*b*ArcTanh[Cos[c + d*x]])/(16*a) - Lo g[256*a^2 - 256*a^2*Cos[c + d*x]^2]/2))/d + (Cot[c + d*x]^2*(8*a + 5*b*Sec [c + d*x]))/(2*d)))/4)/6
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))/cot[(c_.) + (d_.)*(x_)], x_Symbo l] :> Int[(b + a*Sin[c + d*x])/Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x]
Time = 0.62 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(151\) |
| default | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(151\) |
| risch | \(i a x +\frac {2 i a c}{d}+\frac {{\mathrm e}^{i \left (d x +c \right )} \left (33 b \,{\mathrm e}^{10 i \left (d x +c \right )}+144 a \,{\mathrm e}^{9 i \left (d x +c \right )}+5 b \,{\mathrm e}^{8 i \left (d x +c \right )}-288 a \,{\mathrm e}^{7 i \left (d x +c \right )}+90 b \,{\mathrm e}^{6 i \left (d x +c \right )}+544 a \,{\mathrm e}^{5 i \left (d x +c \right )}+90 b \,{\mathrm e}^{4 i \left (d x +c \right )}-288 a \,{\mathrm e}^{3 i \left (d x +c \right )}+5 b \,{\mathrm e}^{2 i \left (d x +c \right )}+144 a \,{\mathrm e}^{i \left (d x +c \right )}+33 b \right )}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{16 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{16 d}\) | \(258\) |
Input:
int(cot(d*x+c)^7*(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/6*cot(d*x+c)^6+1/4*cot(d*x+c)^4-1/2*cot(d*x+c)^2-ln(sin(d*x+c)) )+b*(-1/6/sin(d*x+c)^6*cos(d*x+c)^7+1/24/sin(d*x+c)^4*cos(d*x+c)^7-1/16/si n(d*x+c)^2*cos(d*x+c)^7-1/16*cos(d*x+c)^5-5/48*cos(d*x+c)^3-5/16*cos(d*x+c )-5/16*ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.12 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.82 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {66 \, b \cos \left (d x + c\right )^{5} + 144 \, a \cos \left (d x + c\right )^{4} - 80 \, b \cos \left (d x + c\right )^{3} - 216 \, a \cos \left (d x + c\right )^{2} + 30 \, b \cos \left (d x + c\right ) - 3 \, {\left ({\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a + 5 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a - 5 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="fricas")
Output:
1/96*(66*b*cos(d*x + c)^5 + 144*a*cos(d*x + c)^4 - 80*b*cos(d*x + c)^3 - 2 16*a*cos(d*x + c)^2 + 30*b*cos(d*x + c) - 3*((16*a - 5*b)*cos(d*x + c)^6 - 3*(16*a - 5*b)*cos(d*x + c)^4 + 3*(16*a - 5*b)*cos(d*x + c)^2 - 16*a + 5* b)*log(1/2*cos(d*x + c) + 1/2) - 3*((16*a + 5*b)*cos(d*x + c)^6 - 3*(16*a + 5*b)*cos(d*x + c)^4 + 3*(16*a + 5*b)*cos(d*x + c)^2 - 16*a - 5*b)*log(-1 /2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3* d*cos(d*x + c)^2 - d)
\[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{7}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**7*(a+b*sec(d*x+c)),x)
Output:
Integral((a + b*sec(c + d*x))*cot(c + d*x)**7, x)
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {3 \, {\left (16 \, a - 5 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, {\left (16 \, a + 5 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (33 \, b \cos \left (d x + c\right )^{5} + 72 \, a \cos \left (d x + c\right )^{4} - 40 \, b \cos \left (d x + c\right )^{3} - 108 \, a \cos \left (d x + c\right )^{2} + 15 \, b \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \] Input:
integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="maxima")
Output:
-1/96*(3*(16*a - 5*b)*log(cos(d*x + c) + 1) + 3*(16*a + 5*b)*log(cos(d*x + c) - 1) - 2*(33*b*cos(d*x + c)^5 + 72*a*cos(d*x + c)^4 - 40*b*cos(d*x + c )^3 - 108*a*cos(d*x + c)^2 + 15*b*cos(d*x + c) + 44*a)/(cos(d*x + c)^6 - 3 *cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1))/d
Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.98 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (16 \, a - 5 \, b\right )} \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{32 \, d} - \frac {{\left (16 \, a + 5 \, b\right )} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{32 \, d} + \frac {33 \, b \cos \left (d x + c\right )^{5} + 72 \, a \cos \left (d x + c\right )^{4} - 40 \, b \cos \left (d x + c\right )^{3} - 108 \, a \cos \left (d x + c\right )^{2} + 15 \, b \cos \left (d x + c\right ) + 44 \, a}{48 \, d {\left (\cos \left (d x + c\right ) + 1\right )}^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}} \] Input:
integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="giac")
Output:
-1/32*(16*a - 5*b)*log(abs(cos(d*x + c) + 1))/d - 1/32*(16*a + 5*b)*log(ab s(cos(d*x + c) - 1))/d + 1/48*(33*b*cos(d*x + c)^5 + 72*a*cos(d*x + c)^4 - 40*b*cos(d*x + c)^3 - 108*a*cos(d*x + c)^2 + 15*b*cos(d*x + c) + 44*a)/(d *(cos(d*x + c) + 1)^3*(cos(d*x + c) - 1)^3)
Time = 10.94 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.31 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{32}-\frac {3\,b}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\left (\frac {29\,a}{2}+\frac {15\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-\frac {3\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{6}+\frac {b}{6}\right )}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {29\,a}{128}-\frac {15\,b}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a}{384}-\frac {b}{384}\right )}{d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a+\frac {5\,b}{16}\right )}{d} \] Input:
int(cot(c + d*x)^7*(a + b/cos(c + d*x)),x)
Output:
(tan(c/2 + (d*x)/2)^4*(a/32 - (3*b)/128))/d - (cot(c/2 + (d*x)/2)^6*(a/6 + b/6 - tan(c/2 + (d*x)/2)^2*(2*a + (3*b)/2) + tan(c/2 + (d*x)/2)^4*((29*a) /2 + (15*b)/2)))/(64*d) - (tan(c/2 + (d*x)/2)^2*((29*a)/128 - (15*b)/128)) /d - (tan(c/2 + (d*x)/2)^6*(a/384 - b/384))/d + (a*log(tan(c/2 + (d*x)/2)^ 2 + 1))/d - (log(tan(c/2 + (d*x)/2))*(a + (5*b)/16))/d
Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.23 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {-33 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b +26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b -8 \cos \left (d x +c \right ) b +48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6} a -48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a -15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b +25 \sin \left (d x +c \right )^{6} a -72 \sin \left (d x +c \right )^{4} a +36 \sin \left (d x +c \right )^{2} a -8 a}{48 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^7*(a+b*sec(d*x+c)),x)
Output:
( - 33*cos(c + d*x)*sin(c + d*x)**4*b + 26*cos(c + d*x)*sin(c + d*x)**2*b - 8*cos(c + d*x)*b + 48*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6*a - 4 8*log(tan((c + d*x)/2))*sin(c + d*x)**6*a - 15*log(tan((c + d*x)/2))*sin(c + d*x)**6*b + 25*sin(c + d*x)**6*a - 72*sin(c + d*x)**4*a + 36*sin(c + d* x)**2*a - 8*a)/(48*sin(c + d*x)**6*d)