Integrand size = 21, antiderivative size = 140 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {61 a^4 x}{8}+\frac {2 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \] Output:
-61/8*a^4*x+2*a^4*arctanh(cos(d*x+c))/d+4/3*a^4*cos(d*x+c)^3/d-5*a^4*cot(d *x+c)/d-1/3*a^4*cot(d*x+c)^3/d-2*a^4*cot(d*x+c)*csc(d*x+c)/d-19/8*a^4*cos( d*x+c)*sin(d*x+c)/d-1/4*a^4*cos(d*x+c)*sin(d*x+c)^3/d
Time = 11.51 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (1+\sin (c+d x))^4 \left (-732 (c+d x)+96 \cos (c+d x)+32 \cos (3 (c+d x))-224 \cot \left (\frac {1}{2} (c+d x)\right )-48 \csc ^2\left (\frac {1}{2} (c+d x)\right )+192 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-192 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \sec ^2\left (\frac {1}{2} (c+d x)\right )+32 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+224 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{96 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \] Input:
Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]
Output:
(a^4*(1 + Sin[c + d*x])^4*(-732*(c + d*x) + 96*Cos[c + d*x] + 32*Cos[3*(c + d*x)] - 224*Cot[(c + d*x)/2] - 48*Csc[(c + d*x)/2]^2 + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 48*Sec[(c + d*x)/2]^2 + 32*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 120*Sin[2* (c + d*x)] + 3*Sin[4*(c + d*x)] + 224*Tan[(c + d*x)/2]))/(96*d*(Cos[(c + d *x)/2] + Sin[(c + d*x)/2])^8)
Time = 0.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \cot ^4(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (\csc ^4(c+d x) a^8+\sin ^4(c+d x) a^8+4 \csc ^3(c+d x) a^8+4 \sin ^3(c+d x) a^8+4 \csc ^2(c+d x) a^8+4 \sin ^2(c+d x) a^8-4 \csc (c+d x) a^8-4 \sin (c+d x) a^8-10 a^8\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^8 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^8 \cos ^3(c+d x)}{3 d}-\frac {a^8 \cot ^3(c+d x)}{3 d}-\frac {5 a^8 \cot (c+d x)}{d}-\frac {a^8 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {19 a^8 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {2 a^8 \cot (c+d x) \csc (c+d x)}{d}-\frac {61 a^8 x}{8}}{a^4}\) |
Input:
Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]
Output:
((-61*a^8*x)/8 + (2*a^8*ArcTanh[Cos[c + d*x]])/d + (4*a^8*Cos[c + d*x]^3)/ (3*d) - (5*a^8*Cot[c + d*x])/d - (a^8*Cot[c + d*x]^3)/(3*d) - (2*a^8*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^8*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^8* Cos[c + d*x]*Sin[c + d*x]^3)/(4*d))/a^4
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])
Time = 5.68 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.59
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(222\) |
| default | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(222\) |
| risch | \(-\frac {61 a^{4} x}{8}-\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {5 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {5 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {4 a^{4} \left (-6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+15 i {\mathrm e}^{2 i \left (d x +c \right )}-7 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}\) | \(241\) |
Input:
int(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4*a^ 4*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+6*a^4*(-1/sin(d* x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x-3/2*c)+ 4*a^4*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2*cos(d*x+c)^3-3/2*cos(d*x+c)-3/2* ln(csc(d*x+c)-cot(d*x+c)))+a^4*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c))
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {6 \, a^{4} \cos \left (d x + c\right )^{7} - 75 \, a^{4} \cos \left (d x + c\right )^{5} + 244 \, a^{4} \cos \left (d x + c\right )^{3} - 183 \, a^{4} \cos \left (d x + c\right ) - 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (32 \, a^{4} \cos \left (d x + c\right )^{5} - 183 \, a^{4} d x \cos \left (d x + c\right )^{2} - 32 \, a^{4} \cos \left (d x + c\right )^{3} + 183 \, a^{4} d x + 48 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-1/24*(6*a^4*cos(d*x + c)^7 - 75*a^4*cos(d*x + c)^5 + 244*a^4*cos(d*x + c) ^3 - 183*a^4*cos(d*x + c) - 24*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 24*(a^4*cos(d*x + c)^2 - a^4)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - (32*a^4*cos(d*x + c)^5 - 183*a^4*d*x*cos(d*x + c)^2 - 32*a^4*cos(d*x + c)^3 + 183*a^4*d*x + 48*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
\[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int \cot ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**4*(a+a*sin(d*x+c))**4,x)
Output:
a**4*(Integral(4*sin(c + d*x)*cot(c + d*x)**4, x) + Integral(6*sin(c + d*x )**2*cot(c + d*x)**4, x) + Integral(4*sin(c + d*x)**3*cot(c + d*x)**4, x) + Integral(sin(c + d*x)**4*cot(c + d*x)**4, x) + Integral(cot(c + d*x)**4, x))
Time = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {64 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 288 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 32 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{4} + 96 \, a^{4} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{96 \, d} \] Input:
integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
1/96*(64*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3* log(cos(d*x + c) - 1))*a^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2 *d*x + 2*c))*a^4 - 288*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c) ^3 + tan(d*x + c)))*a^4 + 32*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^4 + 96*a^4*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c ) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (130) = 260\).
Time = 0.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.96 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 183 \, {\left (d x + c\right )} a^{4} - 48 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {88 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/24*(a^4*tan(1/2*d*x + 1/2*c)^3 + 12*a^4*tan(1/2*d*x + 1/2*c)^2 - 183*(d* x + c)*a^4 - 48*a^4*log(abs(tan(1/2*d*x + 1/2*c))) + 57*a^4*tan(1/2*d*x + 1/2*c) + (88*a^4*tan(1/2*d*x + 1/2*c)^3 - 57*a^4*tan(1/2*d*x + 1/2*c)^2 - 12*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2*d*x + 1/2*c)^3 + 2*(57*a^4*tan( 1/2*d*x + 1/2*c)^7 + 96*a^4*tan(1/2*d*x + 1/2*c)^6 + 81*a^4*tan(1/2*d*x + 1/2*c)^5 + 96*a^4*tan(1/2*d*x + 1/2*c)^4 - 81*a^4*tan(1/2*d*x + 1/2*c)^3 + 32*a^4*tan(1/2*d*x + 1/2*c)^2 - 57*a^4*tan(1/2*d*x + 1/2*c) + 32*a^4)/(ta n(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 17.89 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.74 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {2\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {61\,a^4\,\mathrm {atan}\left (\frac {3721\,a^8}{16\,\left (61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {61\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}}\right )}{4\,d}+\frac {19\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-19\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-60\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {67\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-48\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {508\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+116\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {61\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \] Input:
int(cot(c + d*x)^4*(a + a*sin(c + d*x))^4,x)
Output:
(a^4*tan(c/2 + (d*x)/2)^2)/(2*d) + (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (2* a^4*log(tan(c/2 + (d*x)/2)))/d - (61*a^4*atan((3721*a^8)/(16*(61*a^8 - (37 21*a^8*tan(c/2 + (d*x)/2))/16)) + (61*a^8*tan(c/2 + (d*x)/2))/(61*a^8 - (3 721*a^8*tan(c/2 + (d*x)/2))/16)))/(4*d) + (19*a^4*tan(c/2 + (d*x)/2))/(8*d ) - ((61*a^4*tan(c/2 + (d*x)/2)^2)/3 - (16*a^4*tan(c/2 + (d*x)/2)^3)/3 + 1 16*a^4*tan(c/2 + (d*x)/2)^4 + (8*a^4*tan(c/2 + (d*x)/2)^5)/3 + (508*a^4*ta n(c/2 + (d*x)/2)^6)/3 - 48*a^4*tan(c/2 + (d*x)/2)^7 + (67*a^4*tan(c/2 + (d *x)/2)^8)/3 - 60*a^4*tan(c/2 + (d*x)/2)^9 - 19*a^4*tan(c/2 + (d*x)/2)^10 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 32*tan(c/2 + (d*x)/2)^5 + 48*tan(c/2 + (d*x)/2)^7 + 32*tan(c/2 + (d*x)/2)^9 + 8*tan( c/2 + (d*x)/2)^11))
Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-57 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-112 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8 \cos \left (d x +c \right )-48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-183 \sin \left (d x +c \right )^{3} d x +16 \sin \left (d x +c \right )^{3}\right )}{24 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*( - 6*cos(c + d*x)*sin(c + d*x)**6 - 32*cos(c + d*x)*sin(c + d*x)**5 - 57*cos(c + d*x)*sin(c + d*x)**4 + 32*cos(c + d*x)*sin(c + d*x)**3 - 112 *cos(c + d*x)*sin(c + d*x)**2 - 48*cos(c + d*x)*sin(c + d*x) - 8*cos(c + d *x) - 48*log(tan((c + d*x)/2))*sin(c + d*x)**3 - 183*sin(c + d*x)**3*d*x + 16*sin(c + d*x)**3))/(24*sin(c + d*x)**3*d)