Integrand size = 29, antiderivative size = 168 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d} \] Output:
-3/64*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-3/7*a^2*cot(d*x+c)^ 7/d-1/9*a^2*cot(d*x+c)^9/d-3/64*a^2*cot(d*x+c)*csc(d*x+c)/d-1/32*a^2*cot(d *x+c)*csc(d*x+c)^3/d+1/8*a^2*cot(d*x+c)*csc(d*x+c)^5/d-1/4*a^2*cot(d*x+c)^ 3*csc(d*x+c)^5/d
Time = 6.94 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.86 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^9(c+d x) \left (451584 \cos (c+d x)+155904 \cos (3 (c+d x))-20736 \cos (5 (c+d x))-14976 \cos (7 (c+d x))+1664 \cos (9 (c+d x))+119070 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-119070 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+212940 \sin (2 (c+d x))-79380 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+79380 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+195300 \sin (4 (c+d x))+34020 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-34020 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+16380 \sin (6 (c+d x))-8505 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+8505 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-1890 \sin (8 (c+d x))+945 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-945 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{5160960 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]
Output:
-1/5160960*(a^2*Csc[c + d*x]^9*(451584*Cos[c + d*x] + 155904*Cos[3*(c + d* x)] - 20736*Cos[5*(c + d*x)] - 14976*Cos[7*(c + d*x)] + 1664*Cos[9*(c + d* x)] + 119070*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 119070*Log[Sin[(c + d*x) /2]]*Sin[c + d*x] + 212940*Sin[2*(c + d*x)] - 79380*Log[Cos[(c + d*x)/2]]* Sin[3*(c + d*x)] + 79380*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 195300*S in[4*(c + d*x)] + 34020*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 34020*Log [Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16380*Sin[6*(c + d*x)] - 8505*Log[Co s[(c + d*x)/2]]*Sin[7*(c + d*x)] + 8505*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d *x)] - 1890*Sin[8*(c + d*x)] + 945*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 945*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d
Time = 0.53 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^2}{\sin (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \cot ^4(c+d x) \csc ^6(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^5(c+d x)+a^2 \cot ^4(c+d x) \csc ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]
Output:
(-3*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (3* a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) - (3*a^2*Cot[c + d* x]*Csc[c + d*x])/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (a^2* Cot[c + d*x]*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/( 4*d)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 0.70 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(220\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(220\) |
| risch | \(\frac {a^{2} \left (945 \,{\mathrm e}^{17 i \left (d x +c \right )}-330624 i {\mathrm e}^{8 i \left (d x +c \right )}-8190 \,{\mathrm e}^{15 i \left (d x +c \right )}-120960 i {\mathrm e}^{10 i \left (d x +c \right )}-97650 \,{\mathrm e}^{13 i \left (d x +c \right )}+40320 i {\mathrm e}^{14 i \left (d x +c \right )}-106470 \,{\mathrm e}^{11 i \left (d x +c \right )}-8064 i {\mathrm e}^{6 i \left (d x +c \right )}-147840 i {\mathrm e}^{12 i \left (d x +c \right )}+106470 \,{\mathrm e}^{7 i \left (d x +c \right )}+14976 i {\mathrm e}^{2 i \left (d x +c \right )}+97650 \,{\mathrm e}^{5 i \left (d x +c \right )}-19584 i {\mathrm e}^{4 i \left (d x +c \right )}+8190 \,{\mathrm e}^{3 i \left (d x +c \right )}-1664 i-945 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{10080 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}\) | \(238\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-1/7/sin(d*x+c)^7*cos(d*x+c)^5-2/35/sin(d*x+c)^5*cos(d*x+c)^5)+2 *a^2*(-1/8/sin(d*x+c)^8*cos(d*x+c)^5-1/16/sin(d*x+c)^6*cos(d*x+c)^5-1/64/s in(d*x+c)^4*cos(d*x+c)^5+1/128/sin(d*x+c)^2*cos(d*x+c)^5+1/128*cos(d*x+c)^ 3+3/128*cos(d*x+c)+3/128*ln(csc(d*x+c)-cot(d*x+c)))+a^2*(-1/9/sin(d*x+c)^9 *cos(d*x+c)^5-4/63/sin(d*x+c)^7*cos(d*x+c)^5-8/315/sin(d*x+c)^5*cos(d*x+c) ^5))
Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3328 \, a^{2} \cos \left (d x + c\right )^{9} - 14976 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 630 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{7} - 11 \, a^{2} \cos \left (d x + c\right )^{5} - 11 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/40320*(3328*a^2*cos(d*x + c)^9 - 14976*a^2*cos(d*x + c)^7 + 16128*a^2*c os(d*x + c)^5 + 945*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos (d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2)*sin( d*x + c) - 945*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 630*(3*a^2*cos(d*x + c)^7 - 11*a^2*cos(d*x + c)^5 - 11*a^2*cos(d*x + c)^3 + 3*a^2*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^8 - 4*d*cos(d* x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**6*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {315 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1152 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}} - \frac {128 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{40320 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/40320*(315*a^2*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c )^3 + 3*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^ 4 - 4*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1152*(7*tan(d*x + c)^2 + 5)*a^2/tan(d*x + c)^7 - 128*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^2/tan(d*x + c)^9)/d
Time = 0.23 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.55 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15120 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {42774 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 70 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{322560 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/322560*(70*a^2*tan(1/2*d*x + 1/2*c)^9 + 315*a^2*tan(1/2*d*x + 1/2*c)^8 + 450*a^2*tan(1/2*d*x + 1/2*c)^7 - 1008*a^2*tan(1/2*d*x + 1/2*c)^5 - 2520*a ^2*tan(1/2*d*x + 1/2*c)^4 - 3360*a^2*tan(1/2*d*x + 1/2*c)^3 + 15120*a^2*lo g(abs(tan(1/2*d*x + 1/2*c))) + 11340*a^2*tan(1/2*d*x + 1/2*c) - (42774*a^2 *tan(1/2*d*x + 1/2*c)^9 + 11340*a^2*tan(1/2*d*x + 1/2*c)^8 - 3360*a^2*tan( 1/2*d*x + 1/2*c)^6 - 2520*a^2*tan(1/2*d*x + 1/2*c)^5 - 1008*a^2*tan(1/2*d* x + 1/2*c)^4 + 450*a^2*tan(1/2*d*x + 1/2*c)^2 + 315*a^2*tan(1/2*d*x + 1/2* c) + 70*a^2)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 19.70 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.30 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\left (70\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+315\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )}{322560\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^6,x)
Output:
(a^2*(70*sin(c/2 + (d*x)/2)^18 - 70*cos(c/2 + (d*x)/2)^18 + 315*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^17 - 315*cos(c/2 + (d*x)/2)^17*sin(c/2 + (d*x) /2) + 450*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^16 - 1008*cos(c/2 + (d*x )/2)^4*sin(c/2 + (d*x)/2)^14 - 2520*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2 )^13 - 3360*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^12 + 11340*cos(c/2 + ( d*x)/2)^8*sin(c/2 + (d*x)/2)^10 - 11340*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d *x)/2)^8 + 3360*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^6 + 2520*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^5 + 1008*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^4 - 450*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^2 + 15120*log(si n(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/ 2)^9))/(322560*d*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^9)
Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-1664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-945 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-832 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+4416 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+7560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-5040 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2240 \cos \left (d x +c \right )+945 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{9}\right )}{20160 \sin \left (d x +c \right )^{9} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 1664*cos(c + d*x)*sin(c + d*x)**8 - 945*cos(c + d*x)*sin(c + d*x )**7 - 832*cos(c + d*x)*sin(c + d*x)**6 - 630*cos(c + d*x)*sin(c + d*x)**5 + 4416*cos(c + d*x)*sin(c + d*x)**4 + 7560*cos(c + d*x)*sin(c + d*x)**3 + 320*cos(c + d*x)*sin(c + d*x)**2 - 5040*cos(c + d*x)*sin(c + d*x) - 2240* cos(c + d*x) + 945*log(tan((c + d*x)/2))*sin(c + d*x)**9))/(20160*sin(c + d*x)**9*d)