Integrand size = 21, antiderivative size = 259 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {a^3 b^2}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 (a+b)^4 d}-\frac {\left (3 a^2+4 a b-b^2\right ) \log (1+\cos (c+d x))}{16 (a-b)^4 d}+\frac {2 a^3 b \left (a^2+2 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \]
a^3*b^2/(a^2-b^2)^3/d/(b+a*cos(d*x+c))+1/8*(8*a*b*(a^2+b^2)-(3*a^4+12*a^2* b^2+b^4)*cos(d*x+c))*csc(d*x+c)^2/(a^2-b^2)^3/d+1/4*(2*a*b-(a^2+b^2)*cos(d *x+c))*csc(d*x+c)^4/(a^2-b^2)^2/d+1/16*(3*a^2-4*a*b-b^2)*ln(1-cos(d*x+c))/ (a+b)^4/d-1/16*(3*a^2+4*a*b-b^2)*ln(1+cos(d*x+c))/(a-b)^4/d+2*a^3*b*(a^2+2 *b^2)*ln(b+a*cos(d*x+c))/(a^2-b^2)^4/d
Time = 1.62 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {64 a^3 b^2}{(a-b)^3 (a+b)^3}+\frac {2 (-3 a+b) (b+a \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {(b+a \cos (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}+\frac {8 \left (-3 a^2-4 a b+b^2\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}+\frac {128 a^3 b \left (a^2+2 b^2\right ) (b+a \cos (c+d x)) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4}+\frac {8 \left (3 a^2-4 a b-b^2\right ) (b+a \cos (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {2 (3 a+b) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}+\frac {(b+a \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}\right ) \sec ^2(c+d x)}{64 d (a+b \sec (c+d x))^2} \]
((b + a*Cos[c + d*x])*((64*a^3*b^2)/((a - b)^3*(a + b)^3) + (2*(-3*a + b)* (b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^3 - ((b + a*Cos[c + d*x]) *Csc[(c + d*x)/2]^4)/(a + b)^2 + (8*(-3*a^2 - 4*a*b + b^2)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]])/(a - b)^4 + (128*a^3*b*(a^2 + 2*b^2)*(b + a*C os[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4 + (8*(3*a^2 - 4*a*b - b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]])/(a + b)^4 + (2*(3*a + b)* (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^3 + ((b + a*Cos[c + d*x]) *Sec[(c + d*x)/2]^4)/(a - b)^2)*Sec[c + d*x]^2)/(64*d*(a + b*Sec[c + d*x]) ^2)
Time = 1.16 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4360, 3042, 3316, 27, 601, 2178, 2160, 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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{(-a \cos (c+d x)-b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-b\right )^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {a^5 \int \frac {\cos ^2(c+d x)}{(b+a \cos (c+d x))^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^3}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 \int \frac {a^2 \cos ^2(c+d x)}{(b+a \cos (c+d x))^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^3}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {a^3 \left (\frac {2 a^2 b-a \left (a^2+b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\int \frac {-\frac {3 \left (a^2+b^2\right ) \cos ^2(c+d x) a^4}{\left (a^2-b^2\right )^2}+\frac {2 b \left (a^2-3 b^2\right ) \cos (c+d x) a^3}{\left (a^2-b^2\right )^2}+\frac {b^2 \left (a^2+b^2\right ) a^2}{\left (a^2-b^2\right )^2}}{(b+a \cos (c+d x))^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(-a \cos (c+d x))}{4 a^2}\right )}{d}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {a^3 \left (\frac {2 a^2 b-a \left (a^2+b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\frac {\int \frac {-\frac {\left (3 a^4+12 b^2 a^2+b^4\right ) \cos ^2(c+d x) a^4}{\left (a^2-b^2\right )^3}+\frac {2 b \left (5 a^2+b^2\right ) \cos (c+d x) a^3}{\left (a^2-b^2\right )^2}+\frac {b^2 \left (5 a^4+12 b^2 a^2-b^4\right ) a^2}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^2 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(-a \cos (c+d x))}{2 a^2}-\frac {8 a^2 b \left (a^2+b^2\right )-a \left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}\right )}{d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {a^3 \left (\frac {2 a^2 b-a \left (a^2+b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\frac {\int \left (\frac {16 b \left (a^2+2 b^2\right ) a^4}{\left (a^2-b^2\right )^4 (b+a \cos (c+d x))}-\frac {8 b^2 a^4}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^2}-\frac {\left (3 a^2-4 b a-b^2\right ) a}{2 (a+b)^4 (a-a \cos (c+d x))}-\frac {\left (3 a^2+4 b a-b^2\right ) a}{2 (a-b)^4 (\cos (c+d x) a+a)}\right )d(-a \cos (c+d x))}{2 a^2}-\frac {8 a^2 b \left (a^2+b^2\right )-a \left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (\frac {2 a^2 b-a \left (a^2+b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\frac {-\frac {a \left (3 a^2-4 a b-b^2\right ) \log (a-a \cos (c+d x))}{2 (a+b)^4}+\frac {a \left (3 a^2+4 a b-b^2\right ) \log (a \cos (c+d x)+a)}{2 (a-b)^4}-\frac {8 a^4 b^2}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {16 a^4 b \left (a^2+2 b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}}{2 a^2}-\frac {8 a^2 b \left (a^2+b^2\right )-a \left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}\right )}{d}\) |
(a^3*((2*a^2*b - a*(a^2 + b^2)*Cos[c + d*x])/(4*(a^2 - b^2)^2*(a^2 - a^2*C os[c + d*x]^2)^2) - (-1/2*(8*a^2*b*(a^2 + b^2) - a*(3*a^4 + 12*a^2*b^2 + b ^4)*Cos[c + d*x])/((a^2 - b^2)^3*(a^2 - a^2*Cos[c + d*x]^2)) + ((-8*a^4*b^ 2)/((a^2 - b^2)^3*(b + a*Cos[c + d*x])) - (a*(3*a^2 - 4*a*b - b^2)*Log[a - a*Cos[c + d*x]])/(2*(a + b)^4) + (a*(3*a^2 + 4*a*b - b^2)*Log[a + a*Cos[c + d*x]])/(2*(a - b)^4) - (16*a^4*b*(a^2 + 2*b^2)*Log[b + a*Cos[c + d*x]]) /(a^2 - b^2)^4)/(2*a^2))/(4*a^2)))/d
3.3.15.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
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 = 1.47 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} b^{2}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}+\frac {1}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{4}}}{d}\) | \(221\) |
| default | \(\frac {\frac {a^{3} b^{2}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}+\frac {1}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{4}}}{d}\) | \(221\) |
| norman | \(\frac {\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d \left (a -b \right )}+\frac {\left (7 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (a^{6}+18 a^{4} b^{2}+5 a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d \left (a^{7}-b \,a^{6}-3 a^{5} b^{2}+3 a^{4} b^{3}+3 b^{4} a^{3}-3 b^{5} a^{2}-a \,b^{6}+b^{7}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(388\) |
| parallelrisch | \(\frac {8192 b \,a^{3} \left (a^{2}+2 b^{2}\right ) \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+1536 \left (b +a \cos \left (d x +c \right )\right ) \left (a^{2}-\frac {4}{3} a b -\frac {1}{3} b^{2}\right ) \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right ) \left (\left (-16 a^{6}+8 a^{5} b -116 a^{4} b^{2}-28 a^{3} b^{3}-60 a^{2} b^{4}+20 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {31}{2} a^{5} b +\frac {23}{2} a^{3} b^{3}-\frac {53}{2} a^{2} b^{4}+4 a \,b^{5}-\frac {77}{2} a^{4} b^{2}-4 b^{6}-3 a^{6}\right ) \cos \left (3 d x +3 c \right )+\left (6 a^{6}-4 a^{5} b +37 a^{4} b^{2}-a^{3} b^{3}+5 a^{2} b^{4}+5 a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+a^{2} \left (a^{4}-\frac {3}{2} a^{3} b +\frac {39}{2} a^{2} b^{2}+\frac {3}{2} a \,b^{3}+\frac {7}{2} b^{4}\right ) \cos \left (5 d x +5 c \right )+\left (2 a^{6}+49 a^{5} b -13 a^{4} b^{2}-77 a^{3} b^{3}+87 a^{2} b^{4}+28 a \,b^{5}-28 b^{6}\right ) \cos \left (d x +c \right )-22 a^{6}+28 a^{5} b +143 a^{4} b^{2}-35 a^{3} b^{3}+23 a^{2} b^{4}+7 a \,b^{5}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4096 d \left (a -b \right )^{4} \left (a +b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}\) | \(446\) |
| risch | \(\text {Expression too large to display}\) | \(1539\) |
1/d*(a^3*b^2/(a+b)^3/(a-b)^3/(b+a*cos(d*x+c))+2*a^3*b*(a^2+2*b^2)/(a+b)^4/ (a-b)^4*ln(b+a*cos(d*x+c))-1/16/(a+b)^2/(cos(d*x+c)-1)^2-1/16*(-3*a+b)/(a+ b)^3/(cos(d*x+c)-1)+1/16*(3*a^2-4*a*b-b^2)/(a+b)^4*ln(cos(d*x+c)-1)+1/16/( a-b)^2/(cos(d*x+c)+1)^2-1/16*(-3*a-b)/(a-b)^3/(cos(d*x+c)+1)+1/16/(a-b)^4* (-3*a^2-4*a*b+b^2)*ln(cos(d*x+c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1205 vs. \(2 (251) = 502\).
Time = 0.49 (sec) , antiderivative size = 1205, normalized size of antiderivative = 4.65 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
1/16*(40*a^5*b^2 - 32*a^3*b^4 - 8*a*b^6 + 2*(3*a^7 + 17*a^5*b^2 - 19*a^3*b ^4 - a*b^6)*cos(d*x + c)^4 - 2*(5*a^6*b - 9*a^4*b^3 + 3*a^2*b^5 + b^7)*cos (d*x + c)^3 - 2*(5*a^7 + 31*a^5*b^2 - 29*a^3*b^4 - 7*a*b^6)*cos(d*x + c)^2 + 2*(7*a^6*b - 15*a^4*b^3 + 9*a^2*b^5 - b^7)*cos(d*x + c) + 32*(a^5*b^2 + 2*a^3*b^4 + (a^6*b + 2*a^4*b^3)*cos(d*x + c)^5 + (a^5*b^2 + 2*a^3*b^4)*co s(d*x + c)^4 - 2*(a^6*b + 2*a^4*b^3)*cos(d*x + c)^3 - 2*(a^5*b^2 + 2*a^3*b ^4)*cos(d*x + c)^2 + (a^6*b + 2*a^4*b^3)*cos(d*x + c))*log(a*cos(d*x + c) + b) - (3*a^6*b + 16*a^5*b^2 + 33*a^4*b^3 + 32*a^3*b^4 + 13*a^2*b^5 - b^7 + (3*a^7 + 16*a^6*b + 33*a^5*b^2 + 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d* x + c)^5 + (3*a^6*b + 16*a^5*b^2 + 33*a^4*b^3 + 32*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(3*a^7 + 16*a^6*b + 33*a^5*b^2 + 32*a^4*b^3 + 13*a ^3*b^4 - a*b^6)*cos(d*x + c)^3 - 2*(3*a^6*b + 16*a^5*b^2 + 33*a^4*b^3 + 32 *a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^2 + (3*a^7 + 16*a^6*b + 33*a^5*b ^2 + 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (3*a^6*b - 16*a^5*b^2 + 33*a^4*b^3 - 32*a^3*b^4 + 13*a^2*b^5 - b^7 + (3*a^7 - 16*a^6*b + 33*a^5*b^2 - 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d *x + c)^5 + (3*a^6*b - 16*a^5*b^2 + 33*a^4*b^3 - 32*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(3*a^7 - 16*a^6*b + 33*a^5*b^2 - 32*a^4*b^3 + 13* a^3*b^4 - a*b^6)*cos(d*x + c)^3 - 2*(3*a^6*b - 16*a^5*b^2 + 33*a^4*b^3 - 3 2*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^2 + (3*a^7 - 16*a^6*b + 33*a...
\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (251) = 502\).
Time = 0.22 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.97 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {32 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (3 \, a^{2} + 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (20 \, a^{3} b^{2} + 4 \, a b^{4} + {\left (3 \, a^{5} + 20 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{4} b - 4 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{5} + 36 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (7 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )}}{16 \, d} \]
1/16*(32*(a^5*b + 2*a^3*b^3)*log(a*cos(d*x + c) + b)/(a^8 - 4*a^6*b^2 + 6* a^4*b^4 - 4*a^2*b^6 + b^8) - (3*a^2 + 4*a*b - b^2)*log(cos(d*x + c) + 1)/( a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + (3*a^2 - 4*a*b - b^2)*log(cos (d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(20*a^3*b^2 + 4*a*b^4 + (3*a^5 + 20*a^3*b^2 + a*b^4)*cos(d*x + c)^4 - (5*a^4*b - 4*a^ 2*b^3 - b^5)*cos(d*x + c)^3 - (5*a^5 + 36*a^3*b^2 + 7*a*b^4)*cos(d*x + c)^ 2 + (7*a^4*b - 8*a^2*b^3 + b^5)*cos(d*x + c))/(a^6*b - 3*a^4*b^3 + 3*a^2*b ^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^5 + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(a^7 - 3*a^5*b^2 + 3*a^3* b^4 - a*b^6)*cos(d*x + c)^3 - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos( d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (251) = 502\).
Time = 0.40 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.74 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {128 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a^{2} + 2 \, a b + b^{2} - \frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {24 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {128 \, {\left (a^{6} b + a^{4} b^{3} + 2 \, a^{3} b^{4} + \frac {a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}}{64 \, d} \]
1/64*(4*(3*a^2 - 4*a*b - b^2)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 128*(a^5*b + 2*a^3*b^3 )*log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^ 8) - (8*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 8*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 2* a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - b^2*(cos(d*x + c) - 1)^2/( cos(d*x + c) + 1)^2)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (a^2 + 2*a*b + b^2 - 8*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 8*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 18*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 24*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 6*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)^2/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(cos(d*x + c) - 1)^2) - 128*(a^6*b + a^4*b^3 + 2*a^3*b^4 + a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a^5*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c ) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))))/d
Time = 14.70 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.73 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^2}-\frac {5\,b}{8\,{\left (a+b\right )}^3}+\frac {3\,b^2}{8\,{\left (a+b\right )}^4}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{8\,{\left (a-b\right )}^4}+\frac {5\,b}{8\,{\left (a-b\right )}^3}+\frac {3}{16\,{\left (a-b\right )}^2}\right )}{d}+\frac {\frac {{\cos \left (c+d\,x\right )}^4\,\left (3\,a^5+20\,a^3\,b^2+a\,b^4\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\cos \left (c+d\,x\right )\,\left (7\,a^2\,b-b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^3\,\left (5\,a^2\,b+b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (5\,a^5+36\,a^3\,b^2+7\,a\,b^4\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\left (5\,a^3\,b+a\,b^3\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (a\,{\cos \left (c+d\,x\right )}^5+b\,{\cos \left (c+d\,x\right )}^4-2\,a\,{\cos \left (c+d\,x\right )}^3-2\,b\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+b\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^5\,b+4\,a^3\,b^3\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]
(log(cos(c + d*x) - 1)*(3/(16*(a + b)^2) - (5*b)/(8*(a + b)^3) + (3*b^2)/( 8*(a + b)^4)))/d - (log(cos(c + d*x) + 1)*((3*b^2)/(8*(a - b)^4) + (5*b)/( 8*(a - b)^3) + 3/(16*(a - b)^2)))/d + ((cos(c + d*x)^4*(a*b^4 + 3*a^5 + 20 *a^3*b^2))/(8*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (cos(c + d*x)*(7*a^2* b - b^3))/(8*(a^4 + b^4 - 2*a^2*b^2)) - (cos(c + d*x)^3*(5*a^2*b + b^3))/( 8*(a^4 + b^4 - 2*a^2*b^2)) - (cos(c + d*x)^2*(7*a*b^4 + 5*a^5 + 36*a^3*b^2 ))/(8*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (b*(a*b^3 + 5*a^3*b))/(2*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(b + a*cos(c + d*x) - 2*a*cos(c + d*x )^3 + a*cos(c + d*x)^5 - 2*b*cos(c + d*x)^2 + b*cos(c + d*x)^4)) + (log(b + a*cos(c + d*x))*(2*a^5*b + 4*a^3*b^3))/(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4 *b^4 - 4*a^6*b^2))