Integrand size = 21, antiderivative size = 300 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {3 a^3-5 a^2 b+7 a b^2-b^3}{16 (a-b)^2 (a+b)^3 d (1-\cos (c+d x))}+\frac {3 a^3+5 a^2 b+7 a b^2+b^3}{16 (a-b)^3 (a+b)^2 d (1+\cos (c+d x))}+\frac {a^3 b^2}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 (a+b)^4 d}-\frac {\left (3 a^2+4 a b-b^2\right ) \log (1+\cos (c+d x))}{16 (a-b)^4 d}+\frac {2 a^3 b \left (a^2+2 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \] Output:
-1/16*(3*a^3-5*a^2*b+7*a*b^2-b^3)/(a-b)^2/(a+b)^3/d/(1-cos(d*x+c))+1/16*(3 *a^3+5*a^2*b+7*a*b^2+b^3)/(a-b)^3/(a+b)^2/d/(1+cos(d*x+c))+a^3*b^2/(a^2-b^ 2)^3/d/(b+a*cos(d*x+c))+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 = 1.29 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {64 a^3 b^2}{(a-b)^3 (a+b)^3}+\frac {2 (-3 a+b) (b+a \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {(b+a \cos (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}+\frac {8 \left (-3 a^2-4 a b+b^2\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}+\frac {128 a^3 b \left (a^2+2 b^2\right ) (b+a \cos (c+d x)) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4}+\frac {8 \left (3 a^2-4 a b-b^2\right ) (b+a \cos (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {2 (3 a+b) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}+\frac {(b+a \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}\right ) \sec ^2(c+d x)}{64 d (a+b \sec (c+d x))^2} \] Input:
Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]
Output:
((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 = 1.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.99, 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 |
| \(\displaystyle \frac {a^3 \left (\frac {2 a^2 b-a \left (a^2+b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^2 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\frac {-\frac {a \left (3 a^2-4 a b-b^2\right ) \log (a-a \cos (c+d x))}{2 (a+b)^4}+\frac {a \left (3 a^2+4 a b-b^2\right ) \log (a \cos (c+d x)+a)}{2 (a-b)^4}-\frac {8 a^4 b^2}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {16 a^4 b \left (a^2+2 b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}}{2 a^2}-\frac {8 a^2 b \left (a^2+b^2\right )-a \left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]
Output:
(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
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 = 1.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}+\frac {b^{2} a^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(221\) |
| default | \(\frac {\frac {1}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}+\frac {b^{2} a^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(221\) |
| norman | \(\frac {\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d \left (a -b \right )}+\frac {\left (7 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (a^{6}+18 a^{4} b^{2}+5 a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d \left (a^{7}-a^{6} b -3 a^{5} b^{2}+3 a^{4} b^{3}+3 a^{3} b^{4}-3 a^{2} b^{5}-a \,b^{6}+b^{7}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(388\) |
| parallelrisch | \(\frac {8192 b \,a^{3} \left (a^{2}+2 b^{2}\right ) \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+1536 \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right ) \left (a^{2}-\frac {4}{3} a b -\frac {1}{3} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\left (-16 a^{6}+8 a^{5} b -116 a^{4} b^{2}-28 a^{3} b^{3}-60 a^{2} b^{4}+20 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-3 a^{6}-\frac {77}{2} a^{4} b^{2}-\frac {53}{2} a^{2} b^{4}-4 b^{6}-\frac {31}{2} a^{5} b +\frac {23}{2} a^{3} b^{3}+4 a \,b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (6 a^{6}-4 a^{5} b +37 a^{4} b^{2}-a^{3} b^{3}+5 a^{2} b^{4}+5 a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+a^{2} \left (a^{4}-\frac {3}{2} a^{3} b +\frac {39}{2} a^{2} b^{2}+\frac {3}{2} a \,b^{3}+\frac {7}{2} b^{4}\right ) \cos \left (5 d x +5 c \right )+\left (2 a^{6}+49 a^{5} b -13 a^{4} b^{2}-77 a^{3} b^{3}+87 a^{2} b^{4}+28 a \,b^{5}-28 b^{6}\right ) \cos \left (d x +c \right )-22 a^{6}+28 a^{5} b +143 a^{4} b^{2}-35 a^{3} b^{3}+23 a^{2} b^{4}+7 a \,b^{5}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right )}{4096 d \left (a -b \right )^{4} \left (a +b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}\) | \(446\) |
| risch | \(\text {Expression too large to display}\) | \(1539\) |
Input:
int(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/16/(a-b)^2/(1+cos(d*x+c))^2-1/16*(-3*a-b)/(a-b)^3/(1+cos(d*x+c))+1/ 16/(a-b)^4*(-3*a^2-4*a*b+b^2)*ln(1+cos(d*x+c))+b^2*a^3/(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/(-1+cos(d*x+c))^2-1/16*(-3*a+b)/(a+b)^3/(-1+cos(d*x+c))+1/16*(3*a^ 2-4*a*b-b^2)/(a+b)^4*ln(-1+cos(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1205 vs. \(2 (288) = 576\).
Time = 0.30 (sec) , antiderivative size = 1205, normalized size of antiderivative = 4.02 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="fricas")
Output:
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...
\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(csc(d*x+c)**5/(a+b*sec(d*x+c))**2,x)
Output:
Integral(csc(c + d*x)**5/(a + b*sec(c + d*x))**2, x)
Time = 0.05 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.70 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {32 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (3 \, a^{2} + 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (20 \, a^{3} b^{2} + 4 \, a b^{4} + {\left (3 \, a^{5} + 20 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{4} b - 4 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{5} + 36 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (7 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )}}{16 \, d} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="maxima")
Output:
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
Time = 0.16 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.39 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {2 \, {\left (a^{6} b + 2 \, a^{4} b^{3}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{9} d - 4 \, a^{7} b^{2} d + 6 \, a^{5} b^{4} d - 4 \, a^{3} b^{6} d + a b^{8} d} + \frac {{\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left ({\left | -\cos \left (d x + c\right ) + 1 \right |}\right )}{16 \, {\left (a^{4} d + 4 \, a^{3} b d + 6 \, a^{2} b^{2} d + 4 \, a b^{3} d + b^{4} d\right )}} - \frac {{\left (3 \, a^{2} + 4 \, a b - b^{2}\right )} \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{16 \, {\left (a^{4} d - 4 \, a^{3} b d + 6 \, a^{2} b^{2} d - 4 \, a b^{3} d + b^{4} d\right )}} + \frac {20 \, a^{5} b^{2} - 16 \, a^{3} b^{4} - 4 \, a b^{6} + {\left (3 \, a^{7} + 17 \, a^{5} b^{2} - 19 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{6} b - 9 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{7} + 31 \, a^{5} b^{2} - 29 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (7 \, a^{6} b - 15 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )}{8 \, {\left (a \cos \left (d x + c\right ) + b\right )} {\left (a + b\right )}^{4} {\left (a - b\right )}^{4} d {\left (\cos \left (d x + c\right ) + 1\right )}^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="giac")
Output:
2*(a^6*b + 2*a^4*b^3)*log(abs(a*cos(d*x + c) + b))/(a^9*d - 4*a^7*b^2*d + 6*a^5*b^4*d - 4*a^3*b^6*d + a*b^8*d) + 1/16*(3*a^2 - 4*a*b - b^2)*log(abs( -cos(d*x + c) + 1))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/16*(3*a^2 + 4*a*b - b^2)*log(abs(-cos(d*x + c) - 1))/(a^4*d - 4*a^3*b* d + 6*a^2*b^2*d - 4*a*b^3*d + b^4*d) + 1/8*(20*a^5*b^2 - 16*a^3*b^4 - 4*a* b^6 + (3*a^7 + 17*a^5*b^2 - 19*a^3*b^4 - a*b^6)*cos(d*x + c)^4 - (5*a^6*b - 9*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^3 - (5*a^7 + 31*a^5*b^2 - 29*a ^3*b^4 - 7*a*b^6)*cos(d*x + c)^2 + (7*a^6*b - 15*a^4*b^3 + 9*a^2*b^5 - b^7 )*cos(d*x + c))/((a*cos(d*x + c) + b)*(a + b)^4*(a - b)^4*d*(cos(d*x + c) + 1)^2*(cos(d*x + c) - 1)^2)
Time = 11.85 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.49 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^2}-\frac {5\,b}{8\,{\left (a+b\right )}^3}+\frac {3\,b^2}{8\,{\left (a+b\right )}^4}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{8\,{\left (a-b\right )}^4}+\frac {5\,b}{8\,{\left (a-b\right )}^3}+\frac {3}{16\,{\left (a-b\right )}^2}\right )}{d}+\frac {\frac {{\cos \left (c+d\,x\right )}^4\,\left (3\,a^5+20\,a^3\,b^2+a\,b^4\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\cos \left (c+d\,x\right )\,\left (7\,a^2\,b-b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^3\,\left (5\,a^2\,b+b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (5\,a^5+36\,a^3\,b^2+7\,a\,b^4\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\left (5\,a^3\,b+a\,b^3\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (a\,{\cos \left (c+d\,x\right )}^5+b\,{\cos \left (c+d\,x\right )}^4-2\,a\,{\cos \left (c+d\,x\right )}^3-2\,b\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+b\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^5\,b+4\,a^3\,b^3\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \] Input:
int(1/(sin(c + d*x)^5*(a + b/cos(c + d*x))^2),x)
Output:
(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))
Time = 0.19 (sec) , antiderivative size = 1114, normalized size of antiderivative = 3.71 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x)
Output:
(32*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b )*sin(c + d*x)**4*a**6*b + 64*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan ((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*a**4*b**3 + 6*cos(c + d*x)*log (tan((c + d*x)/2))*sin(c + d*x)**4*a**7 - 32*cos(c + d*x)*log(tan((c + d*x )/2))*sin(c + d*x)**4*a**6*b + 66*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**5*b**2 - 64*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)* *4*a**4*b**3 + 26*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**3* b**4 - 2*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a*b**6 - 2*cos (c + d*x)*sin(c + d*x)**4*a**7 - cos(c + d*x)*sin(c + d*x)**4*a**6*b - 36* cos(c + d*x)*sin(c + d*x)**4*a**5*b**2 + 42*cos(c + d*x)*sin(c + d*x)**4*a **4*b**3 - 10*cos(c + d*x)*sin(c + d*x)**4*a**3*b**4 + 7*cos(c + d*x)*sin( c + d*x)**4*a**2*b**5 + 10*cos(c + d*x)*sin(c + d*x)**2*a**6*b - 18*cos(c + d*x)*sin(c + d*x)**2*a**4*b**3 + 6*cos(c + d*x)*sin(c + d*x)**2*a**2*b** 5 + 2*cos(c + d*x)*sin(c + d*x)**2*b**7 + 4*cos(c + d*x)*a**6*b - 12*cos(c + d*x)*a**4*b**3 + 12*cos(c + d*x)*a**2*b**5 - 4*cos(c + d*x)*b**7 + 32*l og(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4* a**5*b**2 + 64*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)* sin(c + d*x)**4*a**3*b**4 + 6*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**6*b - 32*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**5*b**2 + 66*log(tan((c + d* x)/2))*sin(c + d*x)**4*a**4*b**3 - 64*log(tan((c + d*x)/2))*sin(c + d*x...