Integrand size = 21, antiderivative size = 198 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {97 a^4 x}{8}+\frac {5 a^4 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {4 a^4 \cos ^3(c+d x)}{3 d}+\frac {10 a^4 \cot (c+d x)}{d}-\frac {5 a^4 \cot ^3(c+d x)}{3 d}-\frac {a^4 \cot ^5(c+d x)}{5 d}+\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {15 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \] Output:
97/8*a^4*x+5/2*a^4*arctanh(cos(d*x+c))/d-4*a^4*cos(d*x+c)/d-4/3*a^4*cos(d* x+c)^3/d+10*a^4*cot(d*x+c)/d-5/3*a^4*cot(d*x+c)^3/d-1/5*a^4*cot(d*x+c)^5/d +5/2*a^4*cot(d*x+c)*csc(d*x+c)/d-a^4*cot(d*x+c)*csc(d*x+c)^3/d+15/8*a^4*co s(d*x+c)*sin(d*x+c)/d+1/4*a^4*cos(d*x+c)*sin(d*x+c)^3/d
Time = 7.92 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (1+\sin (c+d x))^4 \left (5820 (c+d x)-2400 \cos (c+d x)-160 \cos (3 (c+d x))+2752 \cot \left (\frac {1}{2} (c+d x)\right )+300 \csc ^2\left (\frac {1}{2} (c+d x)\right )-30 \csc ^4\left (\frac {1}{2} (c+d x)\right )+1200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-300 \sec ^2\left (\frac {1}{2} (c+d x)\right )+30 \sec ^4\left (\frac {1}{2} (c+d x)\right )+632 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+96 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )-\frac {79}{2} \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\frac {3}{2} \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+480 \sin (2 (c+d x))-15 \sin (4 (c+d x))-2752 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{480 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \] Input:
Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^4,x]
Output:
(a^4*(1 + Sin[c + d*x])^4*(5820*(c + d*x) - 2400*Cos[c + d*x] - 160*Cos[3* (c + d*x)] + 2752*Cot[(c + d*x)/2] + 300*Csc[(c + d*x)/2]^2 - 30*Csc[(c + d*x)/2]^4 + 1200*Log[Cos[(c + d*x)/2]] - 1200*Log[Sin[(c + d*x)/2]] - 300* Sec[(c + d*x)/2]^2 + 30*Sec[(c + d*x)/2]^4 + 632*Csc[c + d*x]^3*Sin[(c + d *x)/2]^4 + 96*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - (79*Csc[(c + d*x)/2]^4*S in[c + d*x])/2 - (3*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 480*Sin[2*(c + d* x)] - 15*Sin[4*(c + d*x)] - 2752*Tan[(c + d*x)/2]))/(480*d*(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])^8)
Time = 0.64 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (\csc ^6(c+d x) a^{10}+4 \csc ^5(c+d x) a^{10}+3 \csc ^4(c+d x) a^{10}-\sin ^4(c+d x) a^{10}-8 \csc ^3(c+d x) a^{10}-4 \sin ^3(c+d x) a^{10}-14 \csc ^2(c+d x) a^{10}-3 \sin ^2(c+d x) a^{10}+8 \sin (c+d x) a^{10}+14 a^{10}\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {5 a^{10} \text {arctanh}(\cos (c+d x))}{2 d}-\frac {4 a^{10} \cos ^3(c+d x)}{3 d}-\frac {4 a^{10} \cos (c+d x)}{d}-\frac {a^{10} \cot ^5(c+d x)}{5 d}-\frac {5 a^{10} \cot ^3(c+d x)}{3 d}+\frac {10 a^{10} \cot (c+d x)}{d}+\frac {a^{10} \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {15 a^{10} \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^{10} \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {5 a^{10} \cot (c+d x) \csc (c+d x)}{2 d}+\frac {97 a^{10} x}{8}}{a^6}\) |
Input:
Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^4,x]
Output:
((97*a^10*x)/8 + (5*a^10*ArcTanh[Cos[c + d*x]])/(2*d) - (4*a^10*Cos[c + d* x])/d - (4*a^10*Cos[c + d*x]^3)/(3*d) + (10*a^10*Cot[c + d*x])/d - (5*a^10 *Cot[c + d*x]^3)/(3*d) - (a^10*Cot[c + d*x]^5)/(5*d) + (5*a^10*Cot[c + d*x ]*Csc[c + d*x])/(2*d) - (a^10*Cot[c + d*x]*Csc[c + d*x]^3)/d + (15*a^10*Co s[c + d*x]*Sin[c + d*x])/(8*d) + (a^10*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)) /a^6
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Result contains complex when optimal does not.
Time = 11.26 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.54
| method | result | size |
| risch | \(\frac {97 a^{4} x}{8}+\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}-\frac {a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{6 d}-\frac {i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{2 d}-\frac {5 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {5 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 d}-\frac {a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{6 d}-\frac {i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}-\frac {a^{4} \left (-420 i {\mathrm e}^{8 i \left (d x +c \right )}+75 \,{\mathrm e}^{9 i \left (d x +c \right )}+1500 i {\mathrm e}^{6 i \left (d x +c \right )}-30 \,{\mathrm e}^{7 i \left (d x +c \right )}-1940 i {\mathrm e}^{4 i \left (d x +c \right )}+1300 i {\mathrm e}^{2 i \left (d x +c \right )}+30 \,{\mathrm e}^{3 i \left (d x +c \right )}-344 i-75 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(304\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(353\) |
| default | \(\frac {a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(353\) |
Input:
int(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
97/8*a^4*x+1/64*I/d*a^4*exp(4*I*(d*x+c))-1/6*a^4/d*exp(3*I*(d*x+c))-1/2*I/ d*a^4*exp(2*I*(d*x+c))-5/2/d*a^4*exp(I*(d*x+c))-5/2*a^4/d*exp(-I*(d*x+c))+ 1/2*I/d*a^4*exp(-2*I*(d*x+c))-1/6*a^4/d*exp(-3*I*(d*x+c))-1/64*I/d*a^4*exp (-4*I*(d*x+c))-1/15*a^4*(-420*I*exp(8*I*(d*x+c))+75*exp(9*I*(d*x+c))+1500* I*exp(6*I*(d*x+c))-30*exp(7*I*(d*x+c))-1940*I*exp(4*I*(d*x+c))+1300*I*exp( 2*I*(d*x+c))+30*exp(3*I*(d*x+c))-344*I-75*exp(I*(d*x+c)))/d/(exp(2*I*(d*x+ c))-1)^5+5/2*a^4/d*ln(exp(I*(d*x+c))+1)-5/2*a^4/d*ln(exp(I*(d*x+c))-1)
Time = 0.14 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.47 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {30 \, a^{4} \cos \left (d x + c\right )^{9} - 345 \, a^{4} \cos \left (d x + c\right )^{7} + 2231 \, a^{4} \cos \left (d x + c\right )^{5} - 3395 \, a^{4} \cos \left (d x + c\right )^{3} + 1455 \, a^{4} \cos \left (d x + c\right ) + 150 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 150 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 5 \, {\left (32 \, a^{4} \cos \left (d x + c\right )^{7} - 291 \, a^{4} d x \cos \left (d x + c\right )^{4} + 32 \, a^{4} \cos \left (d x + c\right )^{5} + 582 \, a^{4} d x \cos \left (d x + c\right )^{2} - 100 \, a^{4} \cos \left (d x + c\right )^{3} - 291 \, a^{4} d x + 60 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/120*(30*a^4*cos(d*x + c)^9 - 345*a^4*cos(d*x + c)^7 + 2231*a^4*cos(d*x + c)^5 - 3395*a^4*cos(d*x + c)^3 + 1455*a^4*cos(d*x + c) + 150*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 150*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(-1/2*cos( d*x + c) + 1/2)*sin(d*x + c) - 5*(32*a^4*cos(d*x + c)^7 - 291*a^4*d*x*cos( d*x + c)^4 + 32*a^4*cos(d*x + c)^5 + 582*a^4*d*x*cos(d*x + c)^2 - 100*a^4* cos(d*x + c)^3 - 291*a^4*d*x + 60*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos( d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))
\[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int \cot ^{6}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**6*(a+a*sin(d*x+c))**4,x)
Output:
a**4*(Integral(4*sin(c + d*x)*cot(c + d*x)**6, x) + Integral(6*sin(c + d*x )**2*cot(c + d*x)**6, x) + Integral(4*sin(c + d*x)**3*cot(c + d*x)**6, x) + Integral(sin(c + d*x)**4*cot(c + d*x)**6, x) + Integral(cot(c + d*x)**6, x))
Time = 0.12 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {40 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 15 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4} + 8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{4} + 30 \, a^{4} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-1/120*(40*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*co s(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^4 + 15 *(15*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^4 - 120*(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^ 4 + 8*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^4 + 30*a^4*(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) - 1 5*log(cos(d*x + c) - 1)))/d
Time = 0.28 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.71 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 85 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5820 \, {\left (d x + c\right )} a^{4} - 1200 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 2670 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {40 \, {\left (45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 192 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 69 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 384 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 69 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 128 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}} + \frac {2740 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2670 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/480*(3*a^4*tan(1/2*d*x + 1/2*c)^5 + 30*a^4*tan(1/2*d*x + 1/2*c)^4 + 85*a ^4*tan(1/2*d*x + 1/2*c)^3 - 240*a^4*tan(1/2*d*x + 1/2*c)^2 + 5820*(d*x + c )*a^4 - 1200*a^4*log(abs(tan(1/2*d*x + 1/2*c))) - 2670*a^4*tan(1/2*d*x + 1 /2*c) - 40*(45*a^4*tan(1/2*d*x + 1/2*c)^7 + 192*a^4*tan(1/2*d*x + 1/2*c)^6 + 69*a^4*tan(1/2*d*x + 1/2*c)^5 + 384*a^4*tan(1/2*d*x + 1/2*c)^4 - 69*a^4 *tan(1/2*d*x + 1/2*c)^3 + 320*a^4*tan(1/2*d*x + 1/2*c)^2 - 45*a^4*tan(1/2* d*x + 1/2*c) + 128*a^4)/(tan(1/2*d*x + 1/2*c)^2 + 1)^4 + (2740*a^4*tan(1/2 *d*x + 1/2*c)^5 + 2670*a^4*tan(1/2*d*x + 1/2*c)^4 + 240*a^4*tan(1/2*d*x + 1/2*c)^3 - 85*a^4*tan(1/2*d*x + 1/2*c)^2 - 30*a^4*tan(1/2*d*x + 1/2*c) - 3 *a^4)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 17.93 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.29 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^6*(a + a*sin(c + d*x))^4,x)
Output:
(17*a^4*tan(c/2 + (d*x)/2)^3)/(96*d) - (a^4*tan(c/2 + (d*x)/2)^2)/(2*d) + (a^4*tan(c/2 + (d*x)/2)^4)/(16*d) + (a^4*tan(c/2 + (d*x)/2)^5)/(160*d) - ( 5*a^4*log(tan(c/2 + (d*x)/2)))/(2*d) - (97*a^4*atan((9409*a^8)/(16*((485*a ^8)/4 + (9409*a^8*tan(c/2 + (d*x)/2))/16)) - (485*a^8*tan(c/2 + (d*x)/2))/ (4*((485*a^8)/4 + (9409*a^8*tan(c/2 + (d*x)/2))/16))))/(4*d) - ((97*a^4*ta n(c/2 + (d*x)/2)^2)/15 - 8*a^4*tan(c/2 + (d*x)/2)^3 - (2312*a^4*tan(c/2 + (d*x)/2)^4)/15 + (868*a^4*tan(c/2 + (d*x)/2)^5)/3 - (3986*a^4*tan(c/2 + (d *x)/2)^6)/5 + (2296*a^4*tan(c/2 + (d*x)/2)^7)/3 - (18437*a^4*tan(c/2 + (d* x)/2)^8)/15 + 962*a^4*tan(c/2 + (d*x)/2)^9 - (1567*a^4*tan(c/2 + (d*x)/2)^ 10)/3 + 496*a^4*tan(c/2 + (d*x)/2)^11 - 58*a^4*tan(c/2 + (d*x)/2)^12 + a^4 /5 + 2*a^4*tan(c/2 + (d*x)/2))/(d*(32*tan(c/2 + (d*x)/2)^5 + 128*tan(c/2 + (d*x)/2)^7 + 192*tan(c/2 + (d*x)/2)^9 + 128*tan(c/2 + (d*x)/2)^11 + 32*ta n(c/2 + (d*x)/2)^13)) - (89*a^4*tan(c/2 + (d*x)/2))/(16*d)
Time = 0.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.97 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+225 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+1376 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+300 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )-24 \cos \left (d x +c \right )-300 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5}+1455 \sin \left (d x +c \right )^{5} d x +445 \sin \left (d x +c \right )^{5}\right )}{120 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*(30*cos(c + d*x)*sin(c + d*x)**8 + 160*cos(c + d*x)*sin(c + d*x)**7 + 225*cos(c + d*x)*sin(c + d*x)**6 - 640*cos(c + d*x)*sin(c + d*x)**5 + 13 76*cos(c + d*x)*sin(c + d*x)**4 + 300*cos(c + d*x)*sin(c + d*x)**3 - 152*c os(c + d*x)*sin(c + d*x)**2 - 120*cos(c + d*x)*sin(c + d*x) - 24*cos(c + d *x) - 300*log(tan((c + d*x)/2))*sin(c + d*x)**5 + 1455*sin(c + d*x)**5*d*x + 445*sin(c + d*x)**5))/(120*sin(c + d*x)**5*d)