Integrand size = 29, antiderivative size = 212 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}+\frac {b \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac {b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d} \] Output:
b*(a^2-b^2)^2*csc(d*x+c)/a^6/d-1/2*(a^2-b^2)^2*csc(d*x+c)^2/a^5/d-1/3*b*(2 *a^2-b^2)*csc(d*x+c)^3/a^4/d+1/4*(2*a^2-b^2)*csc(d*x+c)^4/a^3/d+1/5*b*csc( d*x+c)^5/a^2/d-1/6*csc(d*x+c)^6/a/d+b^2*(a^2-b^2)^2*ln(sin(d*x+c))/a^7/d-b ^2*(a^2-b^2)^2*ln(a+b*sin(d*x+c))/a^7/d
Time = 2.92 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {60 a b \left (a^2-b^2\right )^2 \csc (c+d x)-30 a^2 \left (a^2-b^2\right )^2 \csc ^2(c+d x)+20 a^3 b \left (-2 a^2+b^2\right ) \csc ^3(c+d x)+15 a^4 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+12 a^5 b \csc ^5(c+d x)-10 a^6 \csc ^6(c+d x)+60 \left (-a^2 b+b^3\right )^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))}{60 a^7 d} \] Input:
Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
(60*a*b*(a^2 - b^2)^2*Csc[c + d*x] - 30*a^2*(a^2 - b^2)^2*Csc[c + d*x]^2 + 20*a^3*b*(-2*a^2 + b^2)*Csc[c + d*x]^3 + 15*a^4*(2*a^2 - b^2)*Csc[c + d*x ]^4 + 12*a^5*b*Csc[c + d*x]^5 - 10*a^6*Csc[c + d*x]^6 + 60*(-(a^2*b) + b^3 )^2*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(60*a^7*d)
Time = 0.46 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3316, 27, 522, 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.
| \(\displaystyle \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{\sin (c+d x)^7 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {\csc ^7(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{a+b \sin (c+d x)}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \int \frac {\csc ^7(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^7 (a+b \sin (c+d x))}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {b^2 \int \left (\frac {\csc ^7(c+d x)}{a b^3}-\frac {\csc ^6(c+d x)}{a^2 b^2}+\frac {\left (b^4-2 a^2 b^2\right ) \csc ^5(c+d x)}{a^3 b^5}+\frac {\left (2 a^2 b^2-b^4\right ) \csc ^4(c+d x)}{a^4 b^4}+\frac {\left (a^2-b^2\right )^2 \csc ^3(c+d x)}{a^5 b^3}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{a^6 b^2}+\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^7 b}-\frac {\left (a^2-b^2\right )^2}{a^7 (a+b \sin (c+d x))}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^2 \left (\frac {\csc ^5(c+d x)}{5 a^2 b}+\frac {\left (a^2-b^2\right )^2 \log (b \sin (c+d x))}{a^7}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7}+\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 b}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 b^2}-\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 b}+\frac {\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 b^2}-\frac {\csc ^6(c+d x)}{6 a b^2}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
(b^2*(((a^2 - b^2)^2*Csc[c + d*x])/(a^6*b) - ((a^2 - b^2)^2*Csc[c + d*x]^2 )/(2*a^5*b^2) - ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^4*b) + ((2*a^2 - b^2)* Csc[c + d*x]^4)/(4*a^3*b^2) + Csc[c + d*x]^5/(5*a^2*b) - Csc[c + d*x]^6/(6 *a*b^2) + ((a^2 - b^2)^2*Log[b*Sin[c + d*x]])/a^7 - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/a^7))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 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]
Time = 1.07 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(-\frac {\csc \left (d x +c \right )^{6}}{6 a d}+\frac {b \csc \left (d x +c \right )^{5}}{5 a^{2} d}+\frac {\csc \left (d x +c \right )^{4}}{2 a d}-\frac {\csc \left (d x +c \right )^{4} b^{2}}{4 d \,a^{3}}-\frac {2 b \csc \left (d x +c \right )^{3}}{3 a^{2} d}+\frac {\csc \left (d x +c \right )^{3} b^{3}}{3 d \,a^{4}}+\frac {\csc \left (d x +c \right )^{2} b^{2}}{d \,a^{3}}-\frac {\csc \left (d x +c \right )^{2} b^{4}}{2 d \,a^{5}}-\frac {\csc \left (d x +c \right )^{2}}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {2 b^{3} \csc \left (d x +c \right )}{d \,a^{4}}+\frac {b^{5} \csc \left (d x +c \right )}{d \,a^{6}}-\frac {b^{2} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{3}}+\frac {2 b^{4} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{5}}-\frac {b^{6} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{7}}\) | \(272\) |
| default | \(-\frac {\csc \left (d x +c \right )^{6}}{6 a d}+\frac {b \csc \left (d x +c \right )^{5}}{5 a^{2} d}+\frac {\csc \left (d x +c \right )^{4}}{2 a d}-\frac {\csc \left (d x +c \right )^{4} b^{2}}{4 d \,a^{3}}-\frac {2 b \csc \left (d x +c \right )^{3}}{3 a^{2} d}+\frac {\csc \left (d x +c \right )^{3} b^{3}}{3 d \,a^{4}}+\frac {\csc \left (d x +c \right )^{2} b^{2}}{d \,a^{3}}-\frac {\csc \left (d x +c \right )^{2} b^{4}}{2 d \,a^{5}}-\frac {\csc \left (d x +c \right )^{2}}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {2 b^{3} \csc \left (d x +c \right )}{d \,a^{4}}+\frac {b^{5} \csc \left (d x +c \right )}{d \,a^{6}}-\frac {b^{2} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{3}}+\frac {2 b^{4} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{5}}-\frac {b^{6} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{7}}\) | \(272\) |
| risch | \(\frac {2 i \left (120 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-90 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-90 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+30 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-15 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+30 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-15 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-90 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+60 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-130 a^{2} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+130 a^{2} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+15 a^{4} b \,{\mathrm e}^{11 i \left (d x +c \right )}+240 a^{2} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-50 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}-240 a^{2} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+35 a^{4} b \,{\mathrm e}^{3 i \left (d x +c \right )}-30 a^{2} b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-35 a^{4} b \,{\mathrm e}^{9 i \left (d x +c \right )}+78 a^{4} b \,{\mathrm e}^{7 i \left (d x +c \right )}-78 a^{4} b \,{\mathrm e}^{5 i \left (d x +c \right )}+30 a^{2} b^{3} {\mathrm e}^{i \left (d x +c \right )}-15 a^{4} b \,{\mathrm e}^{i \left (d x +c \right )}+15 b^{5} {\mathrm e}^{11 i \left (d x +c \right )}-75 b^{5} {\mathrm e}^{9 i \left (d x +c \right )}-150 b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+75 b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-15 b^{5} {\mathrm e}^{i \left (d x +c \right )}+150 b^{5} {\mathrm e}^{7 i \left (d x +c \right )}\right )}{15 d \,a^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d \,a^{3}}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}+\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{7} d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{3} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right ) b^{4}}{a^{5} d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{7} d}\) | \(700\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^2/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/6*csc(d*x+c)^6/a/d+1/5*b*csc(d*x+c)^5/a^2/d+1/2*csc(d*x+c)^4/a/d-1/4/d/ a^3*csc(d*x+c)^4*b^2-2/3*b*csc(d*x+c)^3/a^2/d+1/3/d/a^4*csc(d*x+c)^3*b^3+1 /d/a^3*csc(d*x+c)^2*b^2-1/2/d/a^5*csc(d*x+c)^2*b^4-1/2*csc(d*x+c)^2/a/d+b* csc(d*x+c)/a^2/d-2/d/a^4*b^3*csc(d*x+c)+1/d/a^6*b^5*csc(d*x+c)-1/d*b^2/a^3 *ln(csc(d*x+c)*a+b)+2/d*b^4/a^5*ln(csc(d*x+c)*a+b)-1/d*b^6/a^7*ln(csc(d*x+ c)*a+b)
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (202) = 404\).
Time = 0.11 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {10 \, a^{6} - 45 \, a^{4} b^{2} + 30 \, a^{2} b^{4} + 30 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} - 15 \, {\left (2 \, a^{6} - 7 \, a^{4} b^{2} + 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 60 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (8 \, a^{5} b - 25 \, a^{3} b^{3} + 15 \, a b^{5} + 15 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{4} - 5 \, {\left (4 \, a^{5} b - 11 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{7} d \cos \left (d x + c\right )^{6} - 3 \, a^{7} d \cos \left (d x + c\right )^{4} + 3 \, a^{7} d \cos \left (d x + c\right )^{2} - a^{7} d\right )}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/60*(10*a^6 - 45*a^4*b^2 + 30*a^2*b^4 + 30*(a^6 - 2*a^4*b^2 + a^2*b^4)*co s(d*x + c)^4 - 15*(2*a^6 - 7*a^4*b^2 + 4*a^2*b^4)*cos(d*x + c)^2 - 60*((a^ 4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a ^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c os(d*x + c)^2)*log(b*sin(d*x + c) + a) + 60*((a^4*b^2 - 2*a^2*b^4 + b^6)*c os(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a^4*b^2 - 2*a^2*b^4 + b^6)* cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*log(-1/2*si n(d*x + c)) - 4*(8*a^5*b - 25*a^3*b^3 + 15*a*b^5 + 15*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c)^4 - 5*(4*a^5*b - 11*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^2 )*sin(d*x + c))/(a^7*d*cos(d*x + c)^6 - 3*a^7*d*cos(d*x + c)^4 + 3*a^7*d*c os(d*x + c)^2 - a^7*d)
\[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**2/(a+b*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**5*csc(c + d*x)**2/(a + b*sin(c + d*x)), x)
Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}} - \frac {12 \, a^{4} b \sin \left (d x + c\right ) + 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{5} - 10 \, a^{5} - 30 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (2 \, a^{5} - a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} \sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/60*(60*(a^4*b^2 - 2*a^2*b^4 + b^6)*log(b*sin(d*x + c) + a)/a^7 - 60*(a^ 4*b^2 - 2*a^2*b^4 + b^6)*log(sin(d*x + c))/a^7 - (12*a^4*b*sin(d*x + c) + 60*(a^4*b - 2*a^2*b^3 + b^5)*sin(d*x + c)^5 - 10*a^5 - 30*(a^5 - 2*a^3*b^2 + a*b^4)*sin(d*x + c)^4 - 20*(2*a^4*b - a^2*b^3)*sin(d*x + c)^3 + 15*(2*a ^5 - a^3*b^2)*sin(d*x + c)^2)/(a^6*sin(d*x + c)^6))/d
Time = 0.16 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7} d} - \frac {{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b d} + \frac {12 \, a^{5} b \sin \left (d x + c\right ) - 10 \, a^{6} + 60 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{5} - 30 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{5} b - a^{3} b^{3}\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (2 \, a^{6} - a^{4} b^{2}\right )} \sin \left (d x + c\right )^{2}}{60 \, a^{7} d \sin \left (d x + c\right )^{6}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
(a^4*b^2 - 2*a^2*b^4 + b^6)*log(abs(sin(d*x + c)))/(a^7*d) - (a^4*b^3 - 2* a^2*b^5 + b^7)*log(abs(b*sin(d*x + c) + a))/(a^7*b*d) + 1/60*(12*a^5*b*sin (d*x + c) - 10*a^6 + 60*(a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c)^5 - 30*(a ^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c)^4 - 20*(2*a^5*b - a^3*b^3)*sin(d*x + c)^3 + 15*(2*a^6 - a^4*b^2)*sin(d*x + c)^2)/(a^7*d*sin(d*x + c)^6)
Time = 19.88 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {1}{64\,a}-\frac {b^2}{64\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {b}{96\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{3\,a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b}{32\,a^2}-\frac {2\,b\,\left (\frac {b^2}{16\,a^3}-\frac {5}{64\,a}+\frac {2\,b\,\left (\frac {b}{32\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{a}\right )}{a}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b^2}{32\,a^3}-\frac {5}{128\,a}+\frac {b\,\left (\frac {b}{32\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{a}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a^7\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a^7\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {10\,a^4\,b}{3}-\frac {8\,a^2\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {5\,a^5}{2}-12\,a^3\,b^2+8\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (20\,a^4\,b-56\,a^2\,b^3+32\,b^5\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^5-a^3\,b^2\right )+\frac {a^5}{6}-\frac {2\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}}{64\,a^6\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \] Input:
int(cot(c + d*x)^5/(sin(c + d*x)^2*(a + b*sin(c + d*x))),x)
Output:
(tan(c/2 + (d*x)/2)^4*(1/(64*a) - b^2/(64*a^3)))/d - (tan(c/2 + (d*x)/2)^3 *(b/(96*a^2) + (2*b*(1/(16*a) - b^2/(16*a^3)))/(3*a)))/d - tan(c/2 + (d*x) /2)^6/(384*a*d) + (tan(c/2 + (d*x)/2)*(b/(32*a^2) - (2*b*(b^2/(16*a^3) - 5 /(64*a) + (2*b*(b/(32*a^2) + (2*b*(1/(16*a) - b^2/(16*a^3)))/a))/a))/a + ( 2*b*(1/(16*a) - b^2/(16*a^3)))/a))/d + (tan(c/2 + (d*x)/2)^2*(b^2/(32*a^3) - 5/(128*a) + (b*(b/(32*a^2) + (2*b*(1/(16*a) - b^2/(16*a^3)))/a))/a))/d + (log(tan(c/2 + (d*x)/2))*(b^6 - 2*a^2*b^4 + a^4*b^2))/(a^7*d) + (b*tan(c /2 + (d*x)/2)^5)/(160*a^2*d) - (log(a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2)*(b^6 - 2*a^2*b^4 + a^4*b^2))/(a^7*d) - (tan(c/2 + (d*x)/2)^ 3*((10*a^4*b)/3 - (8*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^4*(8*a*b^4 + (5*a^5) /2 - 12*a^3*b^2) - tan(c/2 + (d*x)/2)^5*(20*a^4*b + 32*b^5 - 56*a^2*b^3) - tan(c/2 + (d*x)/2)^2*(a^5 - a^3*b^2) + a^5/6 - (2*a^4*b*tan(c/2 + (d*x)/2 ))/5)/(64*a^6*d*tan(c/2 + (d*x)/2)^6)
Time = 0.19 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.03 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right )^{6} a^{4} b^{2}+960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right )^{6} a^{2} b^{4}-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right )^{6} b^{6}+480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{4} b^{2}-960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{2} b^{4}+480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b^{6}+55 \sin \left (d x +c \right )^{6} a^{6}-195 \sin \left (d x +c \right )^{6} a^{4} b^{2}+120 \sin \left (d x +c \right )^{6} a^{2} b^{4}+480 \sin \left (d x +c \right )^{5} a^{5} b -960 \sin \left (d x +c \right )^{5} a^{3} b^{3}+480 \sin \left (d x +c \right )^{5} a \,b^{5}-240 \sin \left (d x +c \right )^{4} a^{6}+480 \sin \left (d x +c \right )^{4} a^{4} b^{2}-240 \sin \left (d x +c \right )^{4} a^{2} b^{4}-320 \sin \left (d x +c \right )^{3} a^{5} b +160 \sin \left (d x +c \right )^{3} a^{3} b^{3}+240 \sin \left (d x +c \right )^{2} a^{6}-120 \sin \left (d x +c \right )^{2} a^{4} b^{2}+96 \sin \left (d x +c \right ) a^{5} b -80 a^{6}}{480 \sin \left (d x +c \right )^{6} a^{7} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
( - 480*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x) **6*a**4*b**2 + 960*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)* sin(c + d*x)**6*a**2*b**4 - 480*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x )/2)*b + a)*sin(c + d*x)**6*b**6 + 480*log(tan((c + d*x)/2))*sin(c + d*x)* *6*a**4*b**2 - 960*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**2*b**4 + 480*l og(tan((c + d*x)/2))*sin(c + d*x)**6*b**6 + 55*sin(c + d*x)**6*a**6 - 195* sin(c + d*x)**6*a**4*b**2 + 120*sin(c + d*x)**6*a**2*b**4 + 480*sin(c + d* x)**5*a**5*b - 960*sin(c + d*x)**5*a**3*b**3 + 480*sin(c + d*x)**5*a*b**5 - 240*sin(c + d*x)**4*a**6 + 480*sin(c + d*x)**4*a**4*b**2 - 240*sin(c + d *x)**4*a**2*b**4 - 320*sin(c + d*x)**3*a**5*b + 160*sin(c + d*x)**3*a**3*b **3 + 240*sin(c + d*x)**2*a**6 - 120*sin(c + d*x)**2*a**4*b**2 + 96*sin(c + d*x)*a**5*b - 80*a**6)/(480*sin(c + d*x)**6*a**7*d)