Integrand size = 19, antiderivative size = 140 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {3 b \cot ^2(c+d x)}{2 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {b \log (\tan (c+d x))}{d} \] Output:
-5/16*a*arctanh(cos(d*x+c))/d-3/2*b*cot(d*x+c)^2/d-3/4*b*cot(d*x+c)^4/d-1/ 6*b*cot(d*x+c)^6/d-5/16*a*cot(d*x+c)*csc(d*x+c)/d-5/24*a*cot(d*x+c)*csc(d* x+c)^3/d-1/6*a*cot(d*x+c)*csc(d*x+c)^5/d+b*ln(tan(d*x+c))/d
Time = 0.03 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.66 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {b \csc ^4(c+d x)}{4 d}-\frac {b \csc ^6(c+d x)}{6 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \log (\cos (c+d x))}{d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b \log (\sin (c+d x))}{d}+\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \] Input:
Integrate[Csc[c + d*x]^7*(a + b*Sec[c + d*x]),x]
Output:
(-5*a*Csc[(c + d*x)/2]^2)/(64*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[ (c + d*x)/2]^6)/(384*d) - (b*Csc[c + d*x]^2)/(2*d) - (b*Csc[c + d*x]^4)/(4 *d) - (b*Csc[c + d*x]^6)/(6*d) - (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (b*L og[Cos[c + d*x]])/d + (5*a*Log[Sin[(c + d*x)/2]])/(16*d) + (b*Log[Sin[c + d*x]])/d + (5*a*Sec[(c + d*x)/2]^2)/(64*d) + (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)
Time = 0.80 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.053, Rules used = {3042, 4360, 25, 25, 3042, 25, 3313, 25, 3042, 3100, 243, 49, 2009, 4255, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-b \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\csc ^7(c+d x) \sec (c+d x) (-a \cos (c+d x)-b)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((b+a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^7(c+d x) \sec (c+d x) (a \cos (c+d x)+b)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {b-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \csc ^7(c+d x)dx-b \int -\csc ^7(c+d x) \sec (c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \csc ^7(c+d x)dx+b \int \csc ^7(c+d x) \sec (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \csc (c+d x)^7dx+b \int \csc (c+d x)^7 \sec (c+d x)dx\) |
| \(\Big \downarrow \) 3100 |
| \(\displaystyle a \int \csc (c+d x)^7dx+\frac {b \int \cot ^7(c+d x) \left (\tan ^2(c+d x)+1\right )^3d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle a \int \csc (c+d x)^7dx+\frac {b \int \cot ^4(c+d x) \left (\tan ^2(c+d x)+1\right )^3d\tan ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle a \int \csc (c+d x)^7dx+\frac {b \int \left (\cot ^4(c+d x)+3 \cot ^3(c+d x)+3 \cot ^2(c+d x)+\cot (c+d x)\right )d\tan ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \csc (c+d x)^7dx+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a \left (\frac {5}{6} \int \csc ^5(c+d x)dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{6} \int \csc (c+d x)^5dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^3(c+d x)dx-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \csc (c+d x)^3dx-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {b \left (-\frac {1}{3} \cot ^3(c+d x)-\frac {3}{2} \cot ^2(c+d x)-3 \cot (c+d x)+\log \left (\tan ^2(c+d x)\right )\right )}{2 d}\) |
Input:
Int[Csc[c + d*x]^7*(a + b*Sec[c + d*x]),x]
Output:
a*(-1/6*(Cot[c + d*x]*Csc[c + d*x]^5)/d + (5*(-1/4*(Cot[c + d*x]*Csc[c + d *x]^3)/d + (3*(-1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + d*x])/ (2*d)))/4))/6) + (b*(-3*Cot[c + d*x] - (3*Cot[c + d*x]^2)/2 - Cot[c + d*x] ^3/3 + Log[Tan[c + d*x]^2]))/(2*d)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+b \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
| default | \(\frac {a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+b \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
| parallelrisch | \(\frac {-384 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-384 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (120 a +384 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a -b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-9 a -12 b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-45 a -87 b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a -b \right )+\left (9 a -12 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+45 a -87 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d}\) | \(167\) |
| norman | \(\frac {-\frac {a +b}{384 d}+\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{384 d}+\frac {\left (3 a -4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{128 d}-\frac {\left (3 a +4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128 d}+\frac {\left (15 a -29 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{128 d}-\frac {\left (15 a +29 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{128 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {\left (5 a +16 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(194\) |
| risch | \(\frac {15 a \,{\mathrm e}^{11 i \left (d x +c \right )}+48 b \,{\mathrm e}^{10 i \left (d x +c \right )}-85 a \,{\mathrm e}^{9 i \left (d x +c \right )}-288 b \,{\mathrm e}^{8 i \left (d x +c \right )}+198 a \,{\mathrm e}^{7 i \left (d x +c \right )}+736 b \,{\mathrm e}^{6 i \left (d x +c \right )}+198 a \,{\mathrm e}^{5 i \left (d x +c \right )}-288 b \,{\mathrm e}^{4 i \left (d x +c \right )}-85 a \,{\mathrm e}^{3 i \left (d x +c \right )}+48 b \,{\mathrm e}^{2 i \left (d x +c \right )}+15 a \,{\mathrm e}^{i \left (d x +c \right )}}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(241\) |
Input:
int(csc(d*x+c)^7*(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*((-1/6*csc(d*x+c)^5-5/24*csc(d*x+c)^3-5/16*csc(d*x+c))*cot(d*x+c)+5 /16*ln(csc(d*x+c)-cot(d*x+c)))+b*(-1/6/sin(d*x+c)^6-1/4/sin(d*x+c)^4-1/2/s in(d*x+c)^2+ln(tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (126) = 252\).
Time = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.03 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {30 \, a \cos \left (d x + c\right )^{5} + 48 \, b \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{3} - 120 \, b \cos \left (d x + c\right )^{2} + 66 \, a \cos \left (d x + c\right ) - 96 \, {\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - 3 \, {\left ({\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a + 16 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a - 16 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, b}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(csc(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="fricas")
Output:
1/96*(30*a*cos(d*x + c)^5 + 48*b*cos(d*x + c)^4 - 80*a*cos(d*x + c)^3 - 12 0*b*cos(d*x + c)^2 + 66*a*cos(d*x + c) - 96*(b*cos(d*x + c)^6 - 3*b*cos(d* x + c)^4 + 3*b*cos(d*x + c)^2 - b)*log(-cos(d*x + c)) - 3*((5*a - 16*b)*co s(d*x + c)^6 - 3*(5*a - 16*b)*cos(d*x + c)^4 + 3*(5*a - 16*b)*cos(d*x + c) ^2 - 5*a + 16*b)*log(1/2*cos(d*x + c) + 1/2) + 3*((5*a + 16*b)*cos(d*x + c )^6 - 3*(5*a + 16*b)*cos(d*x + c)^4 + 3*(5*a + 16*b)*cos(d*x + c)^2 - 5*a - 16*b)*log(-1/2*cos(d*x + c) + 1/2) + 88*b)/(d*cos(d*x + c)^6 - 3*d*cos(d *x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**7*(a+b*sec(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {3 \, {\left (5 \, a - 16 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, a + 16 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 96 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{5} + 24 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, b \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) + 44 \, b\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \] Input:
integrate(csc(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="maxima")
Output:
-1/96*(3*(5*a - 16*b)*log(cos(d*x + c) + 1) - 3*(5*a + 16*b)*log(cos(d*x + c) - 1) + 96*b*log(cos(d*x + c)) - 2*(15*a*cos(d*x + c)^5 + 24*b*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 60*b*cos(d*x + c)^2 + 33*a*cos(d*x + c) + 4 4*b)/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1))/d
Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (5 \, a - 16 \, b\right )} \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{32 \, d} + \frac {{\left (5 \, a + 16 \, b\right )} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{32 \, d} - \frac {b \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {15 \, a \cos \left (d x + c\right )^{5} + 24 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, b \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) + 44 \, b}{48 \, d {\left (\cos \left (d x + c\right ) + 1\right )}^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}} \] Input:
integrate(csc(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="giac")
Output:
-1/32*(5*a - 16*b)*log(abs(cos(d*x + c) + 1))/d + 1/32*(5*a + 16*b)*log(ab s(cos(d*x + c) - 1))/d - b*log(abs(cos(d*x + c)))/d + 1/48*(15*a*cos(d*x + c)^5 + 24*b*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 60*b*cos(d*x + c)^2 + 33*a*cos(d*x + c) + 44*b)/(d*(cos(d*x + c) + 1)^3*(cos(d*x + c) - 1)^3)
Time = 11.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^5}{16}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {5\,a\,{\cos \left (c+d\,x\right )}^3}{6}-\frac {5\,b\,{\cos \left (c+d\,x\right )}^2}{4}+\frac {11\,a\,\cos \left (c+d\,x\right )}{16}+\frac {11\,b}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^6-3\,{\cos \left (c+d\,x\right )}^4+3\,{\cos \left (c+d\,x\right )}^2-1\right )}+\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {5\,a}{32}+\frac {b}{2}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {5\,a}{32}-\frac {b}{2}\right )}{d}-\frac {b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \] Input:
int((a + b/cos(c + d*x))/sin(c + d*x)^7,x)
Output:
((11*b)/12 + (11*a*cos(c + d*x))/16 - (5*a*cos(c + d*x)^3)/6 + (5*a*cos(c + d*x)^5)/16 - (5*b*cos(c + d*x)^2)/4 + (b*cos(c + d*x)^4)/2)/(d*(3*cos(c + d*x)^2 - 3*cos(c + d*x)^4 + cos(c + d*x)^6 - 1)) + (log(cos(c + d*x) - 1 )*((5*a)/32 + b/2))/d - (log(cos(c + d*x) + 1)*((5*a)/32 - b/2))/d - (b*lo g(cos(c + d*x)))/d
Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a -10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -8 \cos \left (d x +c \right ) a -48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6} b -48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6} b +15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a +48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b +19 \sin \left (d x +c \right )^{6} b -24 \sin \left (d x +c \right )^{4} b -12 \sin \left (d x +c \right )^{2} b -8 b}{48 \sin \left (d x +c \right )^{6} d} \] Input:
int(csc(d*x+c)^7*(a+b*sec(d*x+c)),x)
Output:
( - 15*cos(c + d*x)*sin(c + d*x)**4*a - 10*cos(c + d*x)*sin(c + d*x)**2*a - 8*cos(c + d*x)*a - 48*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b - 48*l og(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*b + 15*log(tan((c + d*x)/2))*sin( c + d*x)**6*a + 48*log(tan((c + d*x)/2))*sin(c + d*x)**6*b + 19*sin(c + d* x)**6*b - 24*sin(c + d*x)**4*b - 12*sin(c + d*x)**2*b - 8*b)/(48*sin(c + d *x)**6*d)