Integrand size = 27, antiderivative size = 178 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {45 a^3 x}{8}+\frac {45 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \] Output:
45/8*a^3*x+45/8*a^3*arctanh(cos(d*x+c))/d-5*a^3*cos(d*x+c)/d-a^3*cos(d*x+c )^3/d+5*a^3*cot(d*x+c)/d-a^3*cot(d*x+c)^3/d-3/8*a^3*cot(d*x+c)*csc(d*x+c)/ d-1/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d+3/8*a^3*cos(d*x+c)*sin(d*x+c)/d+1/4*a^ 3*cos(d*x+c)*sin(d*x+c)^3/d
Time = 6.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.32 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (360 (c+d x)-368 \cos (c+d x)-16 \cos (3 (c+d x))+192 \cot \left (\frac {1}{2} (c+d x)\right )-6 \csc ^2\left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )+360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+16 \sin (2 (c+d x))-2 \sin (4 (c+d x))-192 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(360*(c + d*x) - 368*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 192*Cot[(c + d*x)/2] - 6*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 360*Log[Cos[(c + d*x)/2]] - 360*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*x) /2]^2 + Sec[(c + d*x)/2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[ (c + d*x)/2]^4*Sin[c + d*x] + 16*Sin[2*(c + d*x)] - 2*Sin[4*(c + d*x)] - 1 92*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.48 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \cos (c+d x) \cot ^5(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^5(c+d x) a^9+3 \csc ^4(c+d x) a^9-\sin ^4(c+d x) a^9-3 \sin ^3(c+d x) a^9-8 \csc ^2(c+d x) a^9-6 \csc (c+d x) a^9+8 \sin (c+d x) a^9+6 a^9\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {45 a^9 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^9 \cos ^3(c+d x)}{d}-\frac {5 a^9 \cos (c+d x)}{d}-\frac {a^9 \cot ^3(c+d x)}{d}+\frac {5 a^9 \cot (c+d x)}{d}+\frac {a^9 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {3 a^9 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^9 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^9 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {45 a^9 x}{8}}{a^6}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
((45*a^9*x)/8 + (45*a^9*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^9*Cos[c + d*x] )/d - (a^9*Cos[c + d*x]^3)/d + (5*a^9*Cot[c + d*x])/d - (a^9*Cot[c + d*x]^ 3)/d - (3*a^9*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^9*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^9*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^9*Cos[c + d* x]*Sin[c + d*x]^3)/(4*d))/a^6
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Result contains complex when optimal does not.
Time = 7.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.64
| method | result | size |
| risch | \(\frac {45 a^{3} x}{8}+\frac {i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}-\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {a^{3} \left (3 \,{\mathrm e}^{7 i \left (d x +c \right )}-11 \,{\mathrm e}^{5 i \left (d x +c \right )}+64 i {\mathrm e}^{6 i \left (d x +c \right )}-11 \,{\mathrm e}^{3 i \left (d x +c \right )}-144 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+128 i {\mathrm e}^{2 i \left (d x +c \right )}-48 i\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {45 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {45 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(292\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(312\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(312\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
45/8*a^3*x+1/64*I*a^3/d*exp(4*I*(d*x+c))-1/8*a^3/d*exp(3*I*(d*x+c))-1/8*I* a^3/d*exp(2*I*(d*x+c))-23/8*a^3/d*exp(I*(d*x+c))-23/8*a^3/d*exp(-I*(d*x+c) )+1/8*I*a^3/d*exp(-2*I*(d*x+c))-1/8*a^3/d*exp(-3*I*(d*x+c))-1/64*I*a^3/d*e xp(-4*I*(d*x+c))+1/4*a^3*(3*exp(7*I*(d*x+c))-11*exp(5*I*(d*x+c))+64*I*exp( 6*I*(d*x+c))-11*exp(3*I*(d*x+c))-144*I*exp(4*I*(d*x+c))+3*exp(I*(d*x+c))+1 28*I*exp(2*I*(d*x+c))-48*I)/d/(exp(2*I*(d*x+c))-1)^4+45/8*a^3/d*ln(exp(I*( d*x+c))+1)-45/8*a^3/d*ln(exp(I*(d*x+c))-1)
Time = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.45 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {16 \, a^{3} \cos \left (d x + c\right )^{7} - 90 \, a^{3} d x \cos \left (d x + c\right )^{4} + 48 \, a^{3} \cos \left (d x + c\right )^{5} + 180 \, a^{3} d x \cos \left (d x + c\right )^{2} - 150 \, a^{3} \cos \left (d x + c\right )^{3} - 90 \, a^{3} d x + 90 \, a^{3} \cos \left (d x + c\right ) - 45 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{7} - 9 \, a^{3} \cos \left (d x + c\right )^{5} + 60 \, a^{3} \cos \left (d x + c\right )^{3} - 45 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/16*(16*a^3*cos(d*x + c)^7 - 90*a^3*d*x*cos(d*x + c)^4 + 48*a^3*cos(d*x + c)^5 + 180*a^3*d*x*cos(d*x + c)^2 - 150*a^3*cos(d*x + c)^3 - 90*a^3*d*x + 90*a^3*cos(d*x + c) - 45*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^ 3)*log(1/2*cos(d*x + c) + 1/2) + 45*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) + 2*(2*a^3*cos(d*x + c)^7 - 9*a^3 *cos(d*x + c)^5 + 60*a^3*cos(d*x + c)^3 - 45*a^3*cos(d*x + c))*sin(d*x + c ))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
\[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**5*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cos(c + d*x)*cot(c + d*x)**5, x) + Integral(3*sin(c + d*x)* cos(c + d*x)*cot(c + d*x)**5, x) + Integral(3*sin(c + d*x)**2*cos(c + d*x) *cot(c + d*x)**5, x) + Integral(sin(c + d*x)**3*cos(c + d*x)*cot(c + d*x)* *5, x))
Time = 0.12 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.51 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} + 2 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/16*(4*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos( d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^3 + 2*(1 5*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 + 2*tan(d*x + c)^3 + tan(d*x + c)))*a^3 - 8*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*a^3 + a ^3*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c) ^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Time = 0.24 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.76 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 360 \, {\left (d x + c\right )} a^{3} - 360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 184 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {250 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 136 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 32 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 552 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 837 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1248 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 736 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 556 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 152 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{4}}}{64 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(a^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^3*ta n(1/2*d*x + 1/2*c)^2 + 360*(d*x + c)*a^3 - 360*a^3*log(abs(tan(1/2*d*x + 1 /2*c))) - 184*a^3*tan(1/2*d*x + 1/2*c) + (250*a^3*tan(1/2*d*x + 1/2*c)^12 + 136*a^3*tan(1/2*d*x + 1/2*c)^11 - 32*a^3*tan(1/2*d*x + 1/2*c)^10 + 552*a ^3*tan(1/2*d*x + 1/2*c)^9 - 837*a^3*tan(1/2*d*x + 1/2*c)^8 + 1248*a^3*tan( 1/2*d*x + 1/2*c)^7 - 1100*a^3*tan(1/2*d*x + 1/2*c)^6 + 736*a^3*tan(1/2*d*x + 1/2*c)^5 - 556*a^3*tan(1/2*d*x + 1/2*c)^4 + 152*a^3*tan(1/2*d*x + 1/2*c )^3 - 12*a^3*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*tan(1/2*d*x + 1/2*c) - a^3)/(t an(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^4)/d
Time = 33.68 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.35 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {45\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {-34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+258\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-138\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {2337\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}-312\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+525\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-184\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {403\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {45\,a^3\,\mathrm {atan}\left (\frac {2025\,a^6}{16\,\left (\frac {2025\,a^6}{16}+\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,\left (\frac {2025\,a^6}{16}+\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}-\frac {23\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \] Input:
int(cos(c + d*x)*cot(c + d*x)^5*(a + a*sin(c + d*x))^3,x)
Output:
(a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(8*d) + (a^3 *tan(c/2 + (d*x)/2)^4)/(64*d) - (45*a^3*log(tan(c/2 + (d*x)/2)))/(8*d) - ( 3*a^3*tan(c/2 + (d*x)/2)^2 - 38*a^3*tan(c/2 + (d*x)/2)^3 + (403*a^3*tan(c/ 2 + (d*x)/2)^4)/2 - 184*a^3*tan(c/2 + (d*x)/2)^5 + 525*a^3*tan(c/2 + (d*x) /2)^6 - 312*a^3*tan(c/2 + (d*x)/2)^7 + (2337*a^3*tan(c/2 + (d*x)/2)^8)/4 - 138*a^3*tan(c/2 + (d*x)/2)^9 + 258*a^3*tan(c/2 + (d*x)/2)^10 - 34*a^3*tan (c/2 + (d*x)/2)^11 + a^3/4 + 2*a^3*tan(c/2 + (d*x)/2))/(d*(16*tan(c/2 + (d *x)/2)^4 + 64*tan(c/2 + (d*x)/2)^6 + 96*tan(c/2 + (d*x)/2)^8 + 64*tan(c/2 + (d*x)/2)^10 + 16*tan(c/2 + (d*x)/2)^12)) - (45*a^3*atan((2025*a^6)/(16*( (2025*a^6)/16 + (2025*a^6*tan(c/2 + (d*x)/2))/16)) - (2025*a^6*tan(c/2 + ( d*x)/2))/(16*((2025*a^6)/16 + (2025*a^6*tan(c/2 + (d*x)/2))/16))))/(4*d) - (23*a^3*tan(c/2 + (d*x)/2))/(8*d)
Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8 \cos \left (d x +c \right )-180 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+180 \sin \left (d x +c \right )^{4} d x +211 \sin \left (d x +c \right )^{4}\right )}{32 \sin \left (d x +c \right )^{4} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^5*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(8*cos(c + d*x)*sin(c + d*x)**7 + 32*cos(c + d*x)*sin(c + d*x)**6 + 12*cos(c + d*x)*sin(c + d*x)**5 - 192*cos(c + d*x)*sin(c + d*x)**4 + 192*c os(c + d*x)*sin(c + d*x)**3 - 12*cos(c + d*x)*sin(c + d*x)**2 - 32*cos(c + d*x)*sin(c + d*x) - 8*cos(c + d*x) - 180*log(tan((c + d*x)/2))*sin(c + d* x)**4 + 180*sin(c + d*x)**4*d*x + 211*sin(c + d*x)**4))/(32*sin(c + d*x)** 4*d)