Integrand size = 29, antiderivative size = 91 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-3 a^3 x+\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d} \] Output:
-3*a^3*x+1/2*a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d-3*a^3*cot(d*x+c)/d -1/3*a^3*cot(d*x+c)^3/d-3/2*a^3*cot(d*x+c)*csc(d*x+c)/d
Time = 0.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.63 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-72 c-72 d x+24 \cos (c+d x)-32 \cot \left (\frac {1}{2} (c+d x)\right )-9 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 \sec ^2\left (\frac {1}{2} (c+d x)\right )+8 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-\frac {1}{2} \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+32 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d} \] Input:
Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(-72*c - 72*d*x + 24*Cos[c + d*x] - 32*Cot[(c + d*x)/2] - 9*Csc[(c + d*x)/2]^2 + 12*Log[Cos[(c + d*x)/2]] - 12*Log[Sin[(c + d*x)/2]] + 9*Sec[(c + d*x)/2]^2 + 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (Csc[(c + d*x)/2]^4*S in[c + d*x])/2 + 32*Tan[(c + d*x)/2]))/(24*d)
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.
| \(\displaystyle \int \cot ^2(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2 (a \sin (c+d x)+a)^3}{\sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^4(c+d x) a^5+3 \csc ^3(c+d x) a^5+2 \csc ^2(c+d x) a^5-2 \csc (c+d x) a^5-\sin (c+d x) a^5-3 a^5\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^5 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^5 \cos (c+d x)}{d}-\frac {a^5 \cot ^3(c+d x)}{3 d}-\frac {3 a^5 \cot (c+d x)}{d}-\frac {3 a^5 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^5 x}{a^2}\) |
Input:
Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(-3*a^5*x + (a^5*ArcTanh[Cos[c + d*x]])/(2*d) + (a^5*Cos[c + d*x])/d - (3* a^5*Cot[c + d*x])/d - (a^5*Cot[c + d*x]^3)/(3*d) - (3*a^5*Cot[c + d*x]*Csc [c + d*x])/(2*d))/a^2
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]))
Time = 0.75 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(125\) |
| default | \(\frac {a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(125\) |
| risch | \(-3 a^{3} x +\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} \left (-12 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+36 i {\mathrm e}^{2 i \left (d x +c \right )}-16 i-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(152\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+3*a^3*(-cot(d*x+c)-d*x-c)+ 3*a^3*(-1/2/sin(d*x+c)^2*cos(d*x+c)^3-1/2*cos(d*x+c)-1/2*ln(csc(d*x+c)-cot (d*x+c)))-1/3*a^3/sin(d*x+c)^3*cos(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (85) = 170\).
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.98 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {32 \, a^{3} \cos \left (d x + c\right )^{3} - 36 \, a^{3} \cos \left (d x + c\right ) - 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (6 \, a^{3} d x \cos \left (d x + c\right )^{2} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 6 \, a^{3} d x - a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/12*(32*a^3*cos(d*x + c)^3 - 36*a^3*cos(d*x + c) - 3*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 6*(6*a^3*d*x*cos(d*x + c )^2 - 2*a^3*cos(d*x + c)^3 - 6*a^3*d*x - a^3*cos(d*x + c))*sin(d*x + c))/( (d*cos(d*x + c)^2 - d)*sin(d*x + c))
\[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cot(c + d*x)**2*csc(c + d*x)**2, x) + Integral(3*sin(c + d* x)*cot(c + d*x)**2*csc(c + d*x)**2, x) + Integral(3*sin(c + d*x)**2*cot(c + d*x)**2*csc(c + d*x)**2, x) + Integral(sin(c + d*x)**3*cot(c + d*x)**2*c sc(c + d*x)**2, x))
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.29 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4 \, a^{3} \cot \left (d x + c\right )^{3} + 36 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{3} - 9 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{3} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/12*(4*a^3*cot(d*x + c)^3 + 36*(d*x + c + 1/tan(d*x + c))*a^3 - 9*a^3*(2 *cos(d*x + c)/(cos(d*x + c)^2 - 1) + log(cos(d*x + c) + 1) - log(cos(d*x + c) - 1)) - 6*a^3*(2*cos(d*x + c) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)))/d
Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, {\left (d x + c\right )} a^{3} - 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {48 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {22 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 + 9*a^3*tan(1/2*d*x + 1/2*c)^2 - 72*(d*x + c)*a^3 - 12*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 33*a^3*tan(1/2*d*x + 1/ 2*c) + 48*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + (22*a^3*tan(1/2*d*x + 1/2*c)^ 3 - 33*a^3*tan(1/2*d*x + 1/2*c)^2 - 9*a^3*tan(1/2*d*x + 1/2*c) - a^3)/tan( 1/2*d*x + 1/2*c)^3)/d
Time = 18.08 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.74 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {6\,a^3\,\mathrm {atan}\left (\frac {36\,a^6}{6\,a^6-36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {6\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^6-36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \] Input:
int((cot(c + d*x)^2*(a + a*sin(c + d*x))^3)/sin(c + d*x)^2,x)
Output:
(3*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(24*d) - ( a^3*log(tan(c/2 + (d*x)/2)))/(2*d) - (6*a^3*atan((36*a^6)/(6*a^6 - 36*a^6* tan(c/2 + (d*x)/2)) + (6*a^6*tan(c/2 + (d*x)/2))/(6*a^6 - 36*a^6*tan(c/2 + (d*x)/2))))/d - ((34*a^3*tan(c/2 + (d*x)/2)^2)/3 - 13*a^3*tan(c/2 + (d*x) /2)^3 + 11*a^3*tan(c/2 + (d*x)/2)^4 + a^3/3 + 3*a^3*tan(c/2 + (d*x)/2))/(d *(8*tan(c/2 + (d*x)/2)^3 + 8*tan(c/2 + (d*x)/2)^5)) + (11*a^3*tan(c/2 + (d *x)/2))/(8*d)
Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-36 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8 \cos \left (d x +c \right )-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-72 \sin \left (d x +c \right )^{3} d x -15 \sin \left (d x +c \right )^{3}\right )}{24 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(24*cos(c + d*x)*sin(c + d*x)**3 - 64*cos(c + d*x)*sin(c + d*x)**2 - 36*cos(c + d*x)*sin(c + d*x) - 8*cos(c + d*x) - 12*log(tan((c + d*x)/2))* sin(c + d*x)**3 - 72*sin(c + d*x)**3*d*x - 15*sin(c + d*x)**3))/(24*sin(c + d*x)**3*d)