Integrand size = 21, antiderivative size = 362 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {3 a^4-9 a^3 b+21 a^2 b^2-11 a b^3+4 b^4}{16 (a-b)^3 (a+b)^4 d (1-\cos (c+d x))}+\frac {3 a^4+9 a^3 b+21 a^2 b^2+11 a b^3+4 b^4}{16 (a-b)^4 (a+b)^3 d (1+\cos (c+d x))}-\frac {a^2 b^3}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 (a+b)^5 d}-\frac {3 a (a+3 b) \log (1+\cos (c+d x))}{16 (a-b)^5 d}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5 d} \] Output:
-1/16*(3*a^4-9*a^3*b+21*a^2*b^2-11*a*b^3+4*b^4)/(a-b)^3/(a+b)^4/d/(1-cos(d *x+c))+1/16*(3*a^4+9*a^3*b+21*a^2*b^2+11*a*b^3+4*b^4)/(a-b)^4/(a+b)^3/d/(1 +cos(d*x+c))-1/2*a^2*b^3/(a^2-b^2)^3/d/(b+a*cos(d*x+c))^2+3*a^2*b^2*(a^2+b ^2)/(a^2-b^2)^4/d/(b+a*cos(d*x+c))+1/4*(b*(3*a^2+b^2)-a*(a^2+3*b^2)*cos(d* x+c))*csc(d*x+c)^4/(a^2-b^2)^3/d+3/16*a*(a-3*b)*ln(1-cos(d*x+c))/(a+b)^5/d -3/16*a*(a+3*b)*ln(1+cos(d*x+c))/(a-b)^5/d+3*a^2*b*(a^4+5*a^2*b^2+2*b^4)*l n(b+a*cos(d*x+c))/(a^2-b^2)^5/d
Result contains complex when optimal does not.
Time = 3.41 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.37 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {32 a^2 b^3}{(-a+b)^3 (a+b)^3}+\frac {192 a^2 (a-i b) (a+i b) b^2 (b+a \cos (c+d x))}{(a-b)^4 (a+b)^4}-\frac {384 i a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (c+d x) (b+a \cos (c+d x))^2}{(a-b)^5 (a+b)^5}-\frac {24 i a (a-3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a+b)^5}+\frac {24 i a (a+3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a-b)^5}+\frac {6 (-a+b) (b+a \cos (c+d x))^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^4}-\frac {(b+a \cos (c+d x))^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {12 a (a+3 b) (b+a \cos (c+d x))^2 \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^5}+\frac {192 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (b+a \cos (c+d x))^2 \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5}+\frac {12 a (a-3 b) (b+a \cos (c+d x))^2 \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^5}+\frac {6 (a+b) (b+a \cos (c+d x))^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^4}+\frac {(b+a \cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}\right ) \sec ^3(c+d x)}{64 d (a+b \sec (c+d x))^3} \] Input:
Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]
Output:
((b + a*Cos[c + d*x])*((32*a^2*b^3)/((-a + b)^3*(a + b)^3) + (192*a^2*(a - I*b)*(a + I*b)*b^2*(b + a*Cos[c + d*x]))/((a - b)^4*(a + b)^4) - ((384*I) *a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^2)/((a - b )^5*(a + b)^5) - ((24*I)*a*(a - 3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d *x])^2)/(a + b)^5 + ((24*I)*a*(a + 3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^2)/(a - b)^5 + (6*(-a + b)*(b + a*Cos[c + d*x])^2*Csc[(c + d*x)/2] ^2)/(a + b)^4 - ((b + a*Cos[c + d*x])^2*Csc[(c + d*x)/2]^4)/(a + b)^3 - (1 2*a*(a + 3*b)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2]^2])/(a - b)^5 + (192*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(b + a*Cos[c + d*x])^2*Log[b + a*Cos[ c + d*x]])/(a^2 - b^2)^5 + (12*a*(a - 3*b)*(b + a*Cos[c + d*x])^2*Log[Sin[ (c + d*x)/2]^2])/(a + b)^5 + (6*(a + b)*(b + a*Cos[c + d*x])^2*Sec[(c + d* x)/2]^2)/(a - b)^4 + ((b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^4)/(a - b)^3 )*Sec[c + d*x]^3)/(64*d*(a + b*Sec[c + d*x])^3)
Time = 1.48 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 4360, 25, 25, 3042, 25, 3316, 25, 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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(-a \cos (c+d x)-b)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(b+a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {a^5 \int -\frac {\cos ^3(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^5 \int \frac {\cos ^3(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \int \frac {a^3 \cos ^3(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle -\frac {a^2 \left (-\frac {\int \frac {-\frac {3 \left (a^2+3 b^2\right ) \cos ^3(c+d x) a^7}{\left (a^2-b^2\right )^3}+\frac {b \left (3 a^2-23 b^2\right ) \cos ^2(c+d x) a^6}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (a^2+3 b^2\right ) a^4}{\left (a^2-b^2\right )^3}+\frac {b^2 \left (3 a^4-3 b^2 a^2-4 b^4\right ) \cos (c+d x) a^3}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{4 a^2}-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {a^2 \left (-\frac {\frac {\int \frac {-\frac {3 \left (a^4+10 b^2 a^2+5 b^4\right ) \cos ^3(c+d x) a^7}{\left (a^2-b^2\right )^4}+\frac {b \left (15 a^4-26 b^2 a^2-37 b^4\right ) \cos ^2(c+d x) a^6}{\left (a^2-b^2\right )^4}+\frac {3 b^2 \left (5 a^4+18 b^2 a^2-7 b^4\right ) \cos (c+d x) a^5}{\left (a^2-b^2\right )^4}+\frac {b^3 \left (5 a^4+34 b^2 a^2+9 b^4\right ) a^4}{\left (a^2-b^2\right )^4}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}+\frac {a^2 \left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^4 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle -\frac {a^2 \left (-\frac {\frac {\int \left (\frac {24 b \left (a^4+5 b^2 a^2+2 b^4\right ) a^4}{\left (a^2-b^2\right )^5 (b+a \cos (c+d x))}-\frac {24 b^2 \left (a^2+b^2\right ) a^4}{\left (a^2-b^2\right )^4 (b+a \cos (c+d x))^2}+\frac {8 b^3 a^4}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^3}-\frac {3 (a-3 b) a^3}{2 (a+b)^5 (a-a \cos (c+d x))}-\frac {3 (a+3 b) a^3}{2 (a-b)^5 (\cos (c+d x) a+a)}\right )d(a \cos (c+d x))}{2 a^2}+\frac {a^2 \left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^4 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{4 a^2}-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}-\frac {\frac {a^2 \left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^4 \left (a^2-a^2 \cos ^2(c+d x)\right )}+\frac {\frac {3 a^3 (a-3 b) \log (a-a \cos (c+d x))}{2 (a+b)^5}-\frac {3 a^3 (a+3 b) \log (a \cos (c+d x)+a)}{2 (a-b)^5}+\frac {24 a^4 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}+\frac {24 a^4 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^5}-\frac {4 a^4 b^3}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}}{2 a^2}}{4 a^2}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]
Output:
-((a^2*(-1/4*(a^2*(b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x]))/((a^2 - b^2)^3*(a^2 - a^2*Cos[c + d*x]^2)^2) - ((a^2*(4*b*(3*a^4 + 8*a^2*b^2 + b ^4) - 3*a*(a^4 + 10*a^2*b^2 + 5*b^4)*Cos[c + d*x]))/(2*(a^2 - b^2)^4*(a^2 - a^2*Cos[c + d*x]^2)) + ((-4*a^4*b^3)/((a^2 - b^2)^3*(b + a*Cos[c + d*x]) ^2) + (24*a^4*b^2*(a^2 + b^2))/((a^2 - b^2)^4*(b + a*Cos[c + d*x])) + (3*a ^3*(a - 3*b)*Log[a - a*Cos[c + d*x]])/(2*(a + b)^5) - (3*a^3*(a + 3*b)*Log [a + a*Cos[c + d*x]])/(2*(a - b)^5) + (24*a^4*b*(a^4 + 5*a^2*b^2 + 2*b^4)* Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^5)/(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 = 1.76 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (1+\cos \left (d x +c \right )\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}-\frac {1}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (-1+\cos \left (d x +c \right )\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{5}}}{d}\) | \(255\) |
| default | \(\frac {\frac {1}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (1+\cos \left (d x +c \right )\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}-\frac {1}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (-1+\cos \left (d x +c \right )\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{5}}}{d}\) | \(255\) |
| norman | \(\frac {-\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{64 d \left (a -b \right )}-\frac {\left (3 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (3 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{32 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (15 a^{7}+459 a^{5} b^{2}+621 a^{3} b^{4}+57 a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 \left (a^{8}-2 a^{7} b -2 a^{6} b^{2}+6 a^{5} b^{3}-6 a^{3} b^{5}+2 a^{2} b^{6}+2 a \,b^{7}-b^{8}\right ) d}-\frac {\left (15 a^{7}+45 a^{6} b +459 a^{5} b^{2}+249 a^{4} b^{3}+621 a^{3} b^{4}+87 a^{2} b^{5}+57 a \,b^{6}+3 b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}+\frac {3 a \left (a -3 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right )}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{10}-5 b^{2} a^{8}+10 a^{6} b^{4}-10 a^{4} b^{6}+5 a^{2} b^{8}-b^{10}\right )}\) | \(519\) |
| parallelrisch | \(\frac {3 b \,a^{2} \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+\frac {3 a \left (a -3 b \right ) \left (a -b \right )^{5} \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\left (5 b^{8}-\frac {5}{16} a^{8}+5 a^{7} b -\frac {313}{16} a^{6} b^{2}-\frac {251}{2} a^{5} b^{3}-\frac {647}{16} a^{4} b^{4}-\frac {359}{2} a^{3} b^{5}-\frac {459}{16} a^{2} b^{6}-24 a \,b^{7}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {17}{12} b^{8}-\frac {5}{8} a^{8}-\frac {14}{3} a^{7} b -\frac {317}{24} a^{6} b^{2}+\frac {83}{3} a^{5} b^{3}-\frac {367}{24} a^{4} b^{4}+\frac {137}{3} a^{3} b^{5}+\frac {89}{24} a^{2} b^{6}+\frac {10}{3} a \,b^{7}\right ) \cos \left (4 d x +4 c \right )-\frac {5 a \left (a +b \right ) \left (a^{6}+\frac {1}{4} a^{5} b +\frac {611}{20} a^{4} b^{2}+\frac {15}{2} a^{3} b^{3}+\frac {467}{10} a^{2} b^{4}-\frac {31}{4} a \,b^{5}+\frac {163}{20} b^{6}\right ) \cos \left (3 d x +3 c \right )}{3}+\left (a^{6}-\frac {3}{4} a^{5} b +\frac {241}{12} a^{4} b^{2}-\frac {7}{6} a^{3} b^{3}+\frac {51}{2} a^{2} b^{4}+\frac {23}{12} a \,b^{5}+\frac {17}{12} b^{6}\right ) a \left (a +b \right ) \cos \left (5 d x +5 c \right )+\frac {5 a^{2} \left (a^{6}+\frac {16}{15} a^{5} b +\frac {443}{15} a^{4} b^{2}+\frac {136}{15} a^{3} b^{3}+\frac {97}{3} a^{2} b^{4}+\frac {8}{3} a \,b^{5}+\frac {17}{15} b^{6}\right ) \cos \left (6 d x +6 c \right )}{16}+\left (-10 a^{8}+\frac {25}{2} a^{7} b +64 a^{6} b^{2}+\frac {25}{2} a^{5} b^{3}+34 a^{4} b^{4}+\frac {139}{2} a^{3} b^{5}+8 a^{2} b^{6}+\frac {3}{2} a \,b^{7}\right ) \cos \left (d x +c \right )+\frac {17 b^{8}}{4}+\frac {5 a^{8}}{8}+63 a^{5} b^{3}+\frac {621 a^{4} b^{4}}{8}+\frac {103 a^{6} b^{2}}{8}+165 a^{3} b^{5}-\frac {59 a^{2} b^{6}}{8}+10 a \,b^{7}+10 a^{7} b \right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right )}{1024}}{\left (a -b \right )^{5} \left (a +b \right )^{5} d \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right )}\) | \(665\) |
| risch | \(\text {Expression too large to display}\) | \(1848\) |
Input:
int(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/16/(a-b)^3/(1+cos(d*x+c))^2-1/16*(-3*a-3*b)/(a-b)^4/(1+cos(d*x+c))- 3/16*a*(a+3*b)/(a-b)^5*ln(1+cos(d*x+c))-1/2*b^3/(a+b)^3*a^2/(a-b)^3/(b+a*c os(d*x+c))^2+3*a^2*b*(a^4+5*a^2*b^2+2*b^4)/(a+b)^5/(a-b)^5*ln(b+a*cos(d*x+ c))+3*a^2*b^2*(a^2+b^2)/(a+b)^4/(a-b)^4/(b+a*cos(d*x+c))-1/16/(a+b)^3/(-1+ cos(d*x+c))^2-1/16*(-3*a+3*b)/(a+b)^4/(-1+cos(d*x+c))+3/16*a*(a-3*b)/(a+b) ^5*ln(-1+cos(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1803 vs. \(2 (348) = 696\).
Time = 0.36 (sec) , antiderivative size = 1803, normalized size of antiderivative = 4.98 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/16*(76*a^6*b^3 + 36*a^4*b^5 - 108*a^2*b^7 - 4*b^9 + 6*(a^9 + 17*a^7*b^2 - 5*a^5*b^4 - 13*a^3*b^6)*cos(d*x + c)^5 - 12*(a^8*b - 9*a^6*b^3 - a^4*b^5 + 9*a^2*b^7)*cos(d*x + c)^4 - 2*(5*a^9 + 98*a^7*b^2 - 12*a^5*b^4 - 98*a^3 *b^6 + 7*a*b^8)*cos(d*x + c)^3 + 8*(2*a^8*b - 25*a^6*b^3 - 3*a^4*b^5 + 25* a^2*b^7 + b^9)*cos(d*x + c)^2 + 2*(55*a^7*b^2 - 9*a^5*b^4 - 51*a^3*b^6 + 5 *a*b^8)*cos(d*x + c) + 48*(a^6*b^3 + 5*a^4*b^5 + 2*a^2*b^7 + (a^8*b + 5*a^ 6*b^3 + 2*a^4*b^5)*cos(d*x + c)^6 + 2*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b^6)*co s(d*x + c)^5 - (2*a^8*b + 9*a^6*b^3 - a^4*b^5 - 2*a^2*b^7)*cos(d*x + c)^4 - 4*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b^6)*cos(d*x + c)^3 + (a^8*b + 3*a^6*b^3 - 8*a^4*b^5 - 4*a^2*b^7)*cos(d*x + c)^2 + 2*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b ^6)*cos(d*x + c))*log(a*cos(d*x + c) + b) - 3*(a^7*b^2 + 8*a^6*b^3 + 25*a^ 5*b^4 + 40*a^4*b^5 + 35*a^3*b^6 + 16*a^2*b^7 + 3*a*b^8 + (a^9 + 8*a^8*b + 25*a^7*b^2 + 40*a^6*b^3 + 35*a^5*b^4 + 16*a^4*b^5 + 3*a^3*b^6)*cos(d*x + c )^6 + 2*(a^8*b + 8*a^7*b^2 + 25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5 + 16*a^3 *b^6 + 3*a^2*b^7)*cos(d*x + c)^5 - (2*a^9 + 16*a^8*b + 49*a^7*b^2 + 72*a^6 *b^3 + 45*a^5*b^4 - 8*a^4*b^5 - 29*a^3*b^6 - 16*a^2*b^7 - 3*a*b^8)*cos(d*x + c)^4 - 4*(a^8*b + 8*a^7*b^2 + 25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5 + 16 *a^3*b^6 + 3*a^2*b^7)*cos(d*x + c)^3 + (a^9 + 8*a^8*b + 23*a^7*b^2 + 24*a^ 6*b^3 - 15*a^5*b^4 - 64*a^4*b^5 - 67*a^3*b^6 - 32*a^2*b^7 - 6*a*b^8)*cos(d *x + c)^2 + 2*(a^8*b + 8*a^7*b^2 + 25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5...
\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csc(d*x+c)**5/(a+b*sec(d*x+c))**3,x)
Output:
Integral(csc(c + d*x)**5/(a + b*sec(c + d*x))**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (348) = 696\).
Time = 0.05 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/16*(48*(a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*log(a*cos(d*x + c) + b)/(a^10 - 5 *a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10) - 3*(a^2 + 3*a*b)*l og(cos(d*x + c) + 1)/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5) + 3*(a^2 - 3*a*b)*log(cos(d*x + c) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) + 2*(38*a^4*b^3 + 56*a^2*b^5 + 2*b^7 + 3*(a^7 + 18*a^5*b^2 + 13*a^3*b^4)*cos(d*x + c)^5 - 6*(a^6*b - 8*a^4*b^3 - 9*a^2* b^5)*cos(d*x + c)^4 - (5*a^7 + 103*a^5*b^2 + 91*a^3*b^4 - 7*a*b^6)*cos(d*x + c)^3 + 4*(2*a^6*b - 23*a^4*b^3 - 26*a^2*b^5 - b^7)*cos(d*x + c)^2 + (55 *a^5*b^2 + 46*a^3*b^4 - 5*a*b^6)*cos(d*x + c))/(a^8*b^2 - 4*a^6*b^4 + 6*a^ 4*b^6 - 4*a^2*b^8 + b^10 + (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2 *b^8)*cos(d*x + c)^6 + 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^ 9)*cos(d*x + c)^5 - (2*a^10 - 9*a^8*b^2 + 16*a^6*b^4 - 14*a^4*b^6 + 6*a^2* b^8 - b^10)*cos(d*x + c)^4 - 4*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)^3 + (a^10 - 6*a^8*b^2 + 14*a^6*b^4 - 16*a^4*b^6 + 9* a^2*b^8 - 2*b^10)*cos(d*x + c)^2 + 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^ 3*b^7 + a*b^9)*cos(d*x + c)))/d
Time = 0.18 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.59 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 \, {\left (a^{7} b + 5 \, a^{5} b^{3} + 2 \, a^{3} b^{5}\right )} \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{11} d - 5 \, a^{9} b^{2} d + 10 \, a^{7} b^{4} d - 10 \, a^{5} b^{6} d + 5 \, a^{3} b^{8} d - a b^{10} d} + \frac {3 \, {\left (a^{2} - 3 \, a b\right )} \log \left ({\left | -\cos \left (d x + c\right ) + 1 \right |}\right )}{16 \, {\left (a^{5} d + 5 \, a^{4} b d + 10 \, a^{3} b^{2} d + 10 \, a^{2} b^{3} d + 5 \, a b^{4} d + b^{5} d\right )}} - \frac {3 \, {\left (a^{2} + 3 \, a b\right )} \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{16 \, {\left (a^{5} d - 5 \, a^{4} b d + 10 \, a^{3} b^{2} d - 10 \, a^{2} b^{3} d + 5 \, a b^{4} d - b^{5} d\right )}} + \frac {3 \, a^{7} \cos \left (d x + c\right )^{5} + 54 \, a^{5} b^{2} \cos \left (d x + c\right )^{5} + 39 \, a^{3} b^{4} \cos \left (d x + c\right )^{5} - 6 \, a^{6} b \cos \left (d x + c\right )^{4} + 48 \, a^{4} b^{3} \cos \left (d x + c\right )^{4} + 54 \, a^{2} b^{5} \cos \left (d x + c\right )^{4} - 5 \, a^{7} \cos \left (d x + c\right )^{3} - 103 \, a^{5} b^{2} \cos \left (d x + c\right )^{3} - 91 \, a^{3} b^{4} \cos \left (d x + c\right )^{3} + 7 \, a b^{6} \cos \left (d x + c\right )^{3} + 8 \, a^{6} b \cos \left (d x + c\right )^{2} - 92 \, a^{4} b^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{2} b^{5} \cos \left (d x + c\right )^{2} - 4 \, b^{7} \cos \left (d x + c\right )^{2} + 55 \, a^{5} b^{2} \cos \left (d x + c\right ) + 46 \, a^{3} b^{4} \cos \left (d x + c\right ) - 5 \, a b^{6} \cos \left (d x + c\right ) + 38 \, a^{4} b^{3} + 56 \, a^{2} b^{5} + 2 \, b^{7}}{8 \, {\left (a^{8} d - 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d - 4 \, a^{2} b^{6} d + b^{8} d\right )} {\left (a \cos \left (d x + c\right )^{3} + b \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - b\right )}^{2}} \] Input:
integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="giac")
Output:
3*(a^7*b + 5*a^5*b^3 + 2*a^3*b^5)*log(abs(-a*cos(d*x + c) - b))/(a^11*d - 5*a^9*b^2*d + 10*a^7*b^4*d - 10*a^5*b^6*d + 5*a^3*b^8*d - a*b^10*d) + 3/16 *(a^2 - 3*a*b)*log(abs(-cos(d*x + c) + 1))/(a^5*d + 5*a^4*b*d + 10*a^3*b^2 *d + 10*a^2*b^3*d + 5*a*b^4*d + b^5*d) - 3/16*(a^2 + 3*a*b)*log(abs(-cos(d *x + c) - 1))/(a^5*d - 5*a^4*b*d + 10*a^3*b^2*d - 10*a^2*b^3*d + 5*a*b^4*d - b^5*d) + 1/8*(3*a^7*cos(d*x + c)^5 + 54*a^5*b^2*cos(d*x + c)^5 + 39*a^3 *b^4*cos(d*x + c)^5 - 6*a^6*b*cos(d*x + c)^4 + 48*a^4*b^3*cos(d*x + c)^4 + 54*a^2*b^5*cos(d*x + c)^4 - 5*a^7*cos(d*x + c)^3 - 103*a^5*b^2*cos(d*x + c)^3 - 91*a^3*b^4*cos(d*x + c)^3 + 7*a*b^6*cos(d*x + c)^3 + 8*a^6*b*cos(d* x + c)^2 - 92*a^4*b^3*cos(d*x + c)^2 - 104*a^2*b^5*cos(d*x + c)^2 - 4*b^7* cos(d*x + c)^2 + 55*a^5*b^2*cos(d*x + c) + 46*a^3*b^4*cos(d*x + c) - 5*a*b ^6*cos(d*x + c) + 38*a^4*b^3 + 56*a^2*b^5 + 2*b^7)/((a^8*d - 4*a^6*b^2*d + 6*a^4*b^4*d - 4*a^2*b^6*d + b^8*d)*(a*cos(d*x + c)^3 + b*cos(d*x + c)^2 - a*cos(d*x + c) - b)^2)
Time = 12.92 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^3}-\frac {15\,b}{16\,{\left (a+b\right )}^4}+\frac {3\,b^2}{4\,{\left (a+b\right )}^5}\right )}{d}+\frac {\frac {19\,a^4\,b^3+28\,a^2\,b^5+b^7}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (-2\,a^6\,b+23\,a^4\,b^3+26\,a^2\,b^5+b^7\right )}{2\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^4\,\left (-a^6\,b+8\,a^4\,b^3+9\,a^2\,b^5\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^5\,\left (a^7+18\,a^5\,b^2+13\,a^3\,b^4\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {a\,\cos \left (c+d\,x\right )\,\left (55\,a^4\,b^2+46\,a^2\,b^4-5\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {a\,{\cos \left (c+d\,x\right )}^3\,\left (5\,a^6+103\,a^4\,b^2+91\,a^2\,b^4-7\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-2\,b^2\right )-{\cos \left (c+d\,x\right )}^4\,\left (2\,a^2-b^2\right )+b^2+a^2\,{\cos \left (c+d\,x\right )}^6+2\,a\,b\,\cos \left (c+d\,x\right )-4\,a\,b\,{\cos \left (c+d\,x\right )}^3+2\,a\,b\,{\cos \left (c+d\,x\right )}^5\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{4\,{\left (a-b\right )}^5}+\frac {15\,b}{16\,{\left (a-b\right )}^4}+\frac {3}{16\,{\left (a-b\right )}^3}\right )}{d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^6\,b+15\,a^4\,b^3+6\,a^2\,b^5\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )} \] Input:
int(1/(sin(c + d*x)^5*(a + b/cos(c + d*x))^3),x)
Output:
(log(cos(c + d*x) - 1)*(3/(16*(a + b)^3) - (15*b)/(16*(a + b)^4) + (3*b^2) /(4*(a + b)^5)))/d + ((b^7 + 28*a^2*b^5 + 19*a^4*b^3)/(4*(a^8 + b^8 - 4*a^ 2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (cos(c + d*x)^2*(b^7 - 2*a^6*b + 26*a^2* b^5 + 23*a^4*b^3))/(2*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + ( 3*cos(c + d*x)^4*(9*a^2*b^5 - a^6*b + 8*a^4*b^3))/(4*(a^8 + b^8 - 4*a^2*b^ 6 + 6*a^4*b^4 - 4*a^6*b^2)) + (3*cos(c + d*x)^5*(a^7 + 13*a^3*b^4 + 18*a^5 *b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (a*cos(c + d* x)*(46*a^2*b^4 - 5*b^6 + 55*a^4*b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^ 4 - 4*a^6*b^2)) - (a*cos(c + d*x)^3*(5*a^6 - 7*b^6 + 91*a^2*b^4 + 103*a^4* b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)))/(d*(cos(c + d*x )^2*(a^2 - 2*b^2) - cos(c + d*x)^4*(2*a^2 - b^2) + b^2 + a^2*cos(c + d*x)^ 6 + 2*a*b*cos(c + d*x) - 4*a*b*cos(c + d*x)^3 + 2*a*b*cos(c + d*x)^5)) - ( log(cos(c + d*x) + 1)*((3*b^2)/(4*(a - b)^5) + (15*b)/(16*(a - b)^4) + 3/( 16*(a - b)^3)))/d + (log(b + a*cos(c + d*x))*(3*a^6*b + 6*a^2*b^5 + 15*a^4 *b^3))/(d*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2))
Time = 0.24 (sec) , antiderivative size = 2107, normalized size of antiderivative = 5.82 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x)
Output:
(384*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**7*b**2 + 1920*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**5*b**4 + 768*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**3*b**6 + 48*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a* *8*b - 384*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**7*b**2 + 1200*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**6*b**3 - 1920*c os(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**5*b**4 + 1680*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4*b**5 - 768*cos(c + d*x)*l og(tan((c + d*x)/2))*sin(c + d*x)**4*a**3*b**6 + 144*cos(c + d*x)*log(tan( (c + d*x)/2))*sin(c + d*x)**4*a**2*b**7 + 24*cos(c + d*x)*sin(c + d*x)**4* a**9 - 30*cos(c + d*x)*sin(c + d*x)**4*a**8*b + 470*cos(c + d*x)*sin(c + d *x)**4*a**7*b**2 - 918*cos(c + d*x)*sin(c + d*x)**4*a**6*b**3 + 1038*cos(c + d*x)*sin(c + d*x)**4*a**5*b**4 - 1242*cos(c + d*x)*sin(c + d*x)**4*a**4 *b**5 + 738*cos(c + d*x)*sin(c + d*x)**4*a**3*b**6 - 114*cos(c + d*x)*sin( c + d*x)**4*a**2*b**7 + 34*cos(c + d*x)*sin(c + d*x)**4*a*b**8 - 8*cos(c + d*x)*sin(c + d*x)**2*a**9 - 32*cos(c + d*x)*sin(c + d*x)**2*a**7*b**2 + 1 44*cos(c + d*x)*sin(c + d*x)**2*a**5*b**4 - 160*cos(c + d*x)*sin(c + d*x)* *2*a**3*b**6 + 56*cos(c + d*x)*sin(c + d*x)**2*a*b**8 - 16*cos(c + d*x)*a* *9 + 64*cos(c + d*x)*a**7*b**2 - 96*cos(c + d*x)*a**5*b**4 + 64*cos(c +...