Integrand size = 21, antiderivative size = 175 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 a^3 x}{2}-\frac {25 a^3 \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:
13/2*a^3*x-25/8*a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d-1/3*a^3*cos(d*x +c)^3/d+5*a^3*cot(d*x+c)/d-2/3*a^3*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c)^5/d+2 3/8*a^3*cot(d*x+c)*csc(d*x+c)/d-3/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d+3/2*a^3* cos(d*x+c)*sin(d*x+c)/d
Time = 8.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (6240 (c+d x)+720 \cos (c+d x)-80 \cos (3 (c+d x))+2624 \cot \left (\frac {1}{2} (c+d x)\right )+690 \csc ^2\left (\frac {1}{2} (c+d x)\right )-45 \csc ^4\left (\frac {1}{2} (c+d x)\right )-3000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-690 \sec ^2\left (\frac {1}{2} (c+d x)\right )+45 \sec ^4\left (\frac {1}{2} (c+d x)\right )+304 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-19 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+720 \sin (2 (c+d x))-2624 \tan \left (\frac {1}{2} (c+d x)\right )+6 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(6240*(c + d*x) + 720*Cos[c + d*x] - 80*Cos[3*(c + d*x)] + 2624*Cot[(c + d*x)/2] + 690*Csc[(c + d*x)/2]^2 - 45*Csc[(c + d* x)/2]^4 - 3000*Log[Cos[(c + d*x)/2]] + 3000*Log[Sin[(c + d*x)/2]] - 690*Se c[(c + d*x)/2]^2 + 45*Sec[(c + d*x)/2]^4 + 304*Csc[c + d*x]^3*Sin[(c + d*x )/2]^4 - 19*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 720*Sin[2*(c + d*x)] - 2624*Tan[(c + d*x)/2] + 6*Sec[(c + d*x)/2]^ 4*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.52 (sec) , antiderivative size = 179, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (\csc ^6(c+d x) a^9+3 \csc ^5(c+d x) a^9-8 \csc ^3(c+d x) a^9-\sin ^3(c+d x) a^9-6 \csc ^2(c+d x) a^9-3 \sin ^2(c+d x) a^9+6 \csc (c+d x) a^9+8 a^9\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {25 a^9 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^9 \cos ^3(c+d x)}{3 d}+\frac {a^9 \cos (c+d x)}{d}-\frac {a^9 \cot ^5(c+d x)}{5 d}-\frac {2 a^9 \cot ^3(c+d x)}{3 d}+\frac {5 a^9 \cot (c+d x)}{d}+\frac {3 a^9 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 a^9 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {23 a^9 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {13 a^9 x}{2}}{a^6}\) |
Input:
Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]
Output:
((13*a^9*x)/2 - (25*a^9*ArcTanh[Cos[c + d*x]])/(8*d) + (a^9*Cos[c + d*x])/ d - (a^9*Cos[c + d*x]^3)/(3*d) + (5*a^9*Cot[c + d*x])/d - (2*a^9*Cot[c + d *x]^3)/(3*d) - (a^9*Cot[c + d*x]^5)/(5*d) + (23*a^9*Cot[c + d*x]*Csc[c + d *x])/(8*d) - (3*a^9*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^9*Cos[c + d* x]*Sin[c + d*x])/(2*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 = 5.13 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.53
| method | result | size |
| risch | \(\frac {13 a^{3} x}{2}-\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {a^{3} \left (-720 i {\mathrm e}^{8 i \left (d x +c \right )}+345 \,{\mathrm e}^{9 i \left (d x +c \right )}+2880 i {\mathrm e}^{6 i \left (d x +c \right )}-330 \,{\mathrm e}^{7 i \left (d x +c \right )}-3680 i {\mathrm e}^{4 i \left (d x +c \right )}+2560 i {\mathrm e}^{2 i \left (d x +c \right )}+330 \,{\mathrm e}^{3 i \left (d x +c \right )}-656 i-345 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {25 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {25 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(268\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(287\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(287\) |
Input:
int(cot(d*x+c)^6*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
13/2*a^3*x-1/24*a^3/d*exp(3*I*(d*x+c))-3/8*I*a^3/d*exp(2*I*(d*x+c))+3/8*a^ 3/d*exp(I*(d*x+c))+3/8*a^3/d*exp(-I*(d*x+c))+3/8*I*a^3/d*exp(-2*I*(d*x+c)) -1/24*a^3/d*exp(-3*I*(d*x+c))-1/60*a^3*(-720*I*exp(8*I*(d*x+c))+345*exp(9* I*(d*x+c))+2880*I*exp(6*I*(d*x+c))-330*exp(7*I*(d*x+c))-3680*I*exp(4*I*(d* x+c))+2560*I*exp(2*I*(d*x+c))+330*exp(3*I*(d*x+c))-656*I-345*exp(I*(d*x+c) ))/d/(exp(2*I*(d*x+c))-1)^5-25/8*a^3/d*ln(exp(I*(d*x+c))+1)+25/8*a^3/d*ln( exp(I*(d*x+c))-1)
Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.59 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {360 \, a^{3} \cos \left (d x + c\right )^{7} - 2392 \, a^{3} \cos \left (d x + c\right )^{5} + 3640 \, a^{3} \cos \left (d x + c\right )^{3} - 1560 \, a^{3} \cos \left (d x + c\right ) + 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 156 \, a^{3} d x \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{5} + 312 \, a^{3} d x \cos \left (d x + c\right )^{2} + 125 \, a^{3} \cos \left (d x + c\right )^{3} - 156 \, a^{3} d x - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-1/240*(360*a^3*cos(d*x + c)^7 - 2392*a^3*cos(d*x + c)^5 + 3640*a^3*cos(d* x + c)^3 - 1560*a^3*cos(d*x + c) + 375*(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)*sin(d*x + c) - 375*(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)*sin(d*x + c) + 10*(8*a^3*cos(d*x + c)^7 - 156*a^3*d*x*cos(d*x + c)^4 - 40*a^3*cos (d*x + c)^5 + 312*a^3*d*x*cos(d*x + c)^2 + 125*a^3*cos(d*x + c)^3 - 156*a^ 3*d*x - 75*a^3*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))
\[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}\, dx + \int \cot ^{6}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**6*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(3*sin(c + d*x)*cot(c + d*x)**6, x) + Integral(3*sin(c + d*x )**2*cot(c + d*x)**6, x) + Integral(sin(c + d*x)**3*cot(c + d*x)**6, x) + Integral(cot(c + d*x)**6, x))
Time = 0.13 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {20 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + 16 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 45 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-1/240*(20*(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^3 - 12 0*(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 + 16*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*t an(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^3 + 45*a^3*(2*(9*cos(d*x + c)^3 - 7*c os(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 1 5*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Time = 0.24 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6240 \, {\left (d x + c\right )} a^{3} + 3000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {320 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac {6850 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*a^3*tan(1/2*d*x + 1/2*c)^4 + 50*a ^3*tan(1/2*d*x + 1/2*c)^3 - 600*a^3*tan(1/2*d*x + 1/2*c)^2 + 6240*(d*x + c )*a^3 + 3000*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2580*a^3*tan(1/2*d*x + 1 /2*c) - 320*(9*a^3*tan(1/2*d*x + 1/2*c)^5 - 12*a^3*tan(1/2*d*x + 1/2*c)^2 - 9*a^3*tan(1/2*d*x + 1/2*c) - 4*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3 - (68 50*a^3*tan(1/2*d*x + 1/2*c)^5 - 2580*a^3*tan(1/2*d*x + 1/2*c)^4 - 600*a^3* tan(1/2*d*x + 1/2*c)^3 + 50*a^3*tan(1/2*d*x + 1/2*c)^2 + 45*a^3*tan(1/2*d* x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 33.48 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.33 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {25\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}+\frac {13\,a^3\,\mathrm {atan}\left (\frac {169\,a^6}{\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}+\frac {-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {769\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {373\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {1744\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {589\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6}+\frac {402\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {a^3}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}-\frac {43\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \] Input:
int(cot(c + d*x)^6*(a + a*sin(c + d*x))^3,x)
Output:
(5*a^3*tan(c/2 + (d*x)/2)^3)/(96*d) - (5*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (3*a^3*tan(c/2 + (d*x)/2)^4)/(64*d) + (a^3*tan(c/2 + (d*x)/2)^5)/(160*d) + (25*a^3*log(tan(c/2 + (d*x)/2)))/(8*d) + (13*a^3*atan((169*a^6)/((325*a^ 6)/4 - 169*a^6*tan(c/2 + (d*x)/2)) + (325*a^6*tan(c/2 + (d*x)/2))/(4*((325 *a^6)/4 - 169*a^6*tan(c/2 + (d*x)/2)))))/d + ((31*a^3*tan(c/2 + (d*x)/2)^3 )/2 - (34*a^3*tan(c/2 + (d*x)/2)^2)/15 + (402*a^3*tan(c/2 + (d*x)/2)^4)/5 + (589*a^3*tan(c/2 + (d*x)/2)^5)/6 + (1744*a^3*tan(c/2 + (d*x)/2)^6)/5 + ( 373*a^3*tan(c/2 + (d*x)/2)^7)/2 + (769*a^3*tan(c/2 + (d*x)/2)^8)/3 + 20*a^ 3*tan(c/2 + (d*x)/2)^9 - 10*a^3*tan(c/2 + (d*x)/2)^10 - a^3/5 - (3*a^3*tan (c/2 + (d*x)/2))/2)/(d*(32*tan(c/2 + (d*x)/2)^5 + 96*tan(c/2 + (d*x)/2)^7 + 96*tan(c/2 + (d*x)/2)^9 + 32*tan(c/2 + (d*x)/2)^11)) - (43*a^3*tan(c/2 + (d*x)/2))/(16*d)
Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+1440 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+5248 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-720 \cos \left (d x +c \right ) \sin \left (d x +c \right )-192 \cos \left (d x +c \right )+3000 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5}+6240 \sin \left (d x +c \right )^{5} d x -2305 \sin \left (d x +c \right )^{5}\right )}{960 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(320*cos(c + d*x)*sin(c + d*x)**7 + 1440*cos(c + d*x)*sin(c + d*x)** 6 + 640*cos(c + d*x)*sin(c + d*x)**5 + 5248*cos(c + d*x)*sin(c + d*x)**4 + 2760*cos(c + d*x)*sin(c + d*x)**3 - 256*cos(c + d*x)*sin(c + d*x)**2 - 72 0*cos(c + d*x)*sin(c + d*x) - 192*cos(c + d*x) + 3000*log(tan((c + d*x)/2) )*sin(c + d*x)**5 + 6240*sin(c + d*x)**5*d*x - 2305*sin(c + d*x)**5))/(960 *sin(c + d*x)**5*d)