Integrand size = 29, antiderivative size = 194 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d} \] Output:
-17/128*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-5/7*a^3*cot(d*x+c )^7/d-1/9*a^3*cot(d*x+c)^9/d-17/128*a^3*cot(d*x+c)*csc(d*x+c)/d+5/64*a^3*c ot(d*x+c)*csc(d*x+c)^3/d-1/6*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d+3/16*a^3*cot( d*x+c)*csc(d*x+c)^5/d-3/8*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d
Time = 6.89 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.61 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^9(c+d x) \left (1161216 \cos (c+d x)+247296 \cos (3 (c+d x))-198144 \cos (5 (c+d x))-71424 \cos (7 (c+d x))+7936 \cos (9 (c+d x))+674730 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-674730 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+669060 \sin (2 (c+d x))-449820 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+449820 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+676620 \sin (4 (c+d x))+192780 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-192780 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-14700 \sin (6 (c+d x))-48195 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+48195 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-10710 \sin (8 (c+d x))+5355 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-5355 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{10321920 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]
Output:
-1/10321920*(a^3*Csc[c + d*x]^9*(1161216*Cos[c + d*x] + 247296*Cos[3*(c + d*x)] - 198144*Cos[5*(c + d*x)] - 71424*Cos[7*(c + d*x)] + 7936*Cos[9*(c + d*x)] + 674730*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 674730*Log[Sin[(c + d *x)/2]]*Sin[c + d*x] + 669060*Sin[2*(c + d*x)] - 449820*Log[Cos[(c + d*x)/ 2]]*Sin[3*(c + d*x)] + 449820*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 676 620*Sin[4*(c + d*x)] + 192780*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 192 780*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 14700*Sin[6*(c + d*x)] - 4819 5*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 48195*Log[Sin[(c + d*x)/2]]*Sin [7*(c + d*x)] - 10710*Sin[8*(c + d*x)] + 5355*Log[Cos[(c + d*x)/2]]*Sin[9* (c + d*x)] - 5355*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d
Time = 0.61 (sec) , antiderivative size = 194, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^4(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+a^3 \cot ^4(c+d x) \csc ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]
Output:
(-17*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - ( 5*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (3*a^3*Cot[c + d*x]*Csc[c + d *x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*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]
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(\frac {a^{3} \left (5355 \,{\mathrm e}^{17 i \left (d x +c \right )}-1080576 i {\mathrm e}^{8 i \left (d x +c \right )}+7350 \,{\mathrm e}^{15 i \left (d x +c \right )}-80640 i {\mathrm e}^{10 i \left (d x +c \right )}-338310 \,{\mathrm e}^{13 i \left (d x +c \right )}+241920 i {\mathrm e}^{14 i \left (d x +c \right )}-334530 \,{\mathrm e}^{11 i \left (d x +c \right )}+209664 i {\mathrm e}^{6 i \left (d x +c \right )}-456960 i {\mathrm e}^{12 i \left (d x +c \right )}+334530 \,{\mathrm e}^{7 i \left (d x +c \right )}+71424 i {\mathrm e}^{2 i \left (d x +c \right )}+338310 \,{\mathrm e}^{5 i \left (d x +c \right )}-43776 i {\mathrm e}^{4 i \left (d x +c \right )}-7350 \,{\mathrm e}^{3 i \left (d x +c \right )}-7936 i-5355 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{20160 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}+\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}\) | \(238\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(316\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(316\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/20160*a^3*(5355*exp(17*I*(d*x+c))-1080576*I*exp(8*I*(d*x+c))+7350*exp(15 *I*(d*x+c))-80640*I*exp(10*I*(d*x+c))-338310*exp(13*I*(d*x+c))+241920*I*ex p(14*I*(d*x+c))-334530*exp(11*I*(d*x+c))+209664*I*exp(6*I*(d*x+c))-456960* I*exp(12*I*(d*x+c))+334530*exp(7*I*(d*x+c))+71424*I*exp(2*I*(d*x+c))+33831 0*exp(5*I*(d*x+c))-43776*I*exp(4*I*(d*x+c))-7350*exp(3*I*(d*x+c))-7936*I-5 355*exp(I*(d*x+c)))/d/(exp(2*I*(d*x+c))-1)^9-17/128*a^3/d*ln(exp(I*(d*x+c) )+1)+17/128*a^3/d*ln(exp(I*(d*x+c))-1)
Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15872 \, a^{3} \cos \left (d x + c\right )^{9} - 71424 \, a^{3} \cos \left (d x + c\right )^{7} + 64512 \, a^{3} \cos \left (d x + c\right )^{5} + 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 210 \, {\left (51 \, a^{3} \cos \left (d x + c\right )^{7} - 59 \, a^{3} \cos \left (d x + c\right )^{5} - 187 \, a^{3} \cos \left (d x + c\right )^{3} + 51 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/80640*(15872*a^3*cos(d*x + c)^9 - 71424*a^3*cos(d*x + c)^7 + 64512*a^3* cos(d*x + c)^5 + 5355*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*c os(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*si n(d*x + c) - 5355*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d *x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d *x + c) - 210*(51*a^3*cos(d*x + c)^7 - 59*a^3*cos(d*x + c)^5 - 187*a^3*cos (d*x + c)^3 + 51*a^3*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. \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**6*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.38 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {945 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 840 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {6912 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {256 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{80640 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/80640*(945*a^3*(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)) + 840*a^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/( cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 6912*(7*tan(d*x + c)^2 + 5)*a^3/tan (d*x + c)^7 - 256*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d
Time = 0.24 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.68 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {140 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 85680 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 52920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {242386 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 52920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 5040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4032 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 140 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{645120 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/645120*(140*a^3*tan(1/2*d*x + 1/2*c)^9 + 945*a^3*tan(1/2*d*x + 1/2*c)^8 + 2340*a^3*tan(1/2*d*x + 1/2*c)^7 + 1680*a^3*tan(1/2*d*x + 1/2*c)^6 - 4032 *a^3*tan(1/2*d*x + 1/2*c)^5 - 12600*a^3*tan(1/2*d*x + 1/2*c)^4 - 16800*a^3 *tan(1/2*d*x + 1/2*c)^3 - 5040*a^3*tan(1/2*d*x + 1/2*c)^2 + 85680*a^3*log( abs(tan(1/2*d*x + 1/2*c))) + 52920*a^3*tan(1/2*d*x + 1/2*c) - (242386*a^3* tan(1/2*d*x + 1/2*c)^9 + 52920*a^3*tan(1/2*d*x + 1/2*c)^8 - 5040*a^3*tan(1 /2*d*x + 1/2*c)^7 - 16800*a^3*tan(1/2*d*x + 1/2*c)^6 - 12600*a^3*tan(1/2*d *x + 1/2*c)^5 - 4032*a^3*tan(1/2*d*x + 1/2*c)^4 + 1680*a^3*tan(1/2*d*x + 1 /2*c)^3 + 2340*a^3*tan(1/2*d*x + 1/2*c)^2 + 945*a^3*tan(1/2*d*x + 1/2*c) + 140*a^3)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 18.37 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.84 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {17\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {21\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^6,x)
Output:
(a^3*cot(c/2 + (d*x)/2)^2)/(128*d) + (5*a^3*cot(c/2 + (d*x)/2)^3)/(192*d) + (5*a^3*cot(c/2 + (d*x)/2)^4)/(256*d) + (a^3*cot(c/2 + (d*x)/2)^5)/(160*d ) - (a^3*cot(c/2 + (d*x)/2)^6)/(384*d) - (13*a^3*cot(c/2 + (d*x)/2)^7)/(35 84*d) - (3*a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^3*cot(c/2 + (d*x)/2)^9) /(4608*d) - (a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (5*a^3*tan(c/2 + (d*x)/2) ^3)/(192*d) - (5*a^3*tan(c/2 + (d*x)/2)^4)/(256*d) - (a^3*tan(c/2 + (d*x)/ 2)^5)/(160*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) + (13*a^3*tan(c/2 + (d* x)/2)^7)/(3584*d) + (3*a^3*tan(c/2 + (d*x)/2)^8)/(2048*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(4608*d) + (17*a^3*log(tan(c/2 + (d*x)/2)))/(128*d) - (21*a^3 *cot(c/2 + (d*x)/2))/(256*d) + (21*a^3*tan(c/2 + (d*x)/2))/(256*d)
Time = 0.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.88 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-7936 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-5355 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-3968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+9870 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+27264 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+15960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-10880 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-15120 \cos \left (d x +c \right ) \sin \left (d x +c \right )-4480 \cos \left (d x +c \right )+5355 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{9}\right )}{40320 \sin \left (d x +c \right )^{9} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 7936*cos(c + d*x)*sin(c + d*x)**8 - 5355*cos(c + d*x)*sin(c + d* x)**7 - 3968*cos(c + d*x)*sin(c + d*x)**6 + 9870*cos(c + d*x)*sin(c + d*x) **5 + 27264*cos(c + d*x)*sin(c + d*x)**4 + 15960*cos(c + d*x)*sin(c + d*x) **3 - 10880*cos(c + d*x)*sin(c + d*x)**2 - 15120*cos(c + d*x)*sin(c + d*x) - 4480*cos(c + d*x) + 5355*log(tan((c + d*x)/2))*sin(c + d*x)**9))/(40320 *sin(c + d*x)**9*d)