Integrand size = 29, antiderivative size = 73 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-a^2 x+\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d} \] Output:
-a^2*x+a^2*arctanh(cos(d*x+c))/d-a^2*cot(d*x+c)/d-1/3*a^2*cot(d*x+c)^3/d-a ^2*cot(d*x+c)*csc(d*x+c)/d
Time = 0.61 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.92 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (24 c+24 d x+8 \cot \left (\frac {1}{2} (c+d x)\right )+6 \csc ^2\left (\frac {1}{2} (c+d x)\right )-24 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 \sec ^2\left (\frac {1}{2} (c+d x)\right )-8 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+\frac {1}{2} \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-8 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d} \] Input:
Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]
Output:
-1/24*(a^2*(24*c + 24*d*x + 8*Cot[(c + d*x)/2] + 6*Csc[(c + d*x)/2]^2 - 24 *Log[Cos[(c + d*x)/2]] + 24*Log[Sin[(c + d*x)/2]] - 6*Sec[(c + d*x)/2]^2 - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 8*Tan[(c + d*x)/2]))/d
Time = 0.41 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.
\(\displaystyle \int \cot ^2(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^2 (a \sin (c+d x)+a)^2}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3352 |
\(\displaystyle \int \left (a^2 \cot ^2(c+d x)+a^2 \cot ^2(c+d x) \csc ^2(c+d x)+2 a^2 \cot ^2(c+d x) \csc (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}-a^2 x\) |
Input:
Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]
Output:
-(a^2*x) + (a^2*ArcTanh[Cos[c + d*x]])/d - (a^2*Cot[c + d*x])/d - (a^2*Cot [c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{2} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(97\) |
default | \(\frac {a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{2} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(97\) |
risch | \(-a^{2} x +\frac {2 a^{2} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+6 i {\mathrm e}^{2 i \left (d x +c \right )}-2 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(105\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-cot(d*x+c)-d*x-c)+2*a^2*(-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^2/sin(d*x+c)^3*cos(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (71) = 142\).
Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.27 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {4 \, a^{2} \cos \left (d x + c\right )^{3} - 6 \, a^{2} \cos \left (d x + c\right ) - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (a^{2} d x \cos \left (d x + c\right )^{2} - a^{2} d x - a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/6*(4*a^2*cos(d*x + c)^3 - 6*a^2*cos(d*x + c) - 3*(a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(a^2*cos(d*x + c)^2 - a^ 2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 6*(a^2*d*x*cos(d*x + c)^2 - a^2*d*x - a^2*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
\[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**2*(a+a*sin(d*x+c))**2,x)
Output:
a**2*(Integral(cot(c + d*x)**2*csc(c + d*x)**2, x) + Integral(2*sin(c + d* x)*cot(c + d*x)**2*csc(c + d*x)**2, x) + Integral(sin(c + d*x)**2*cot(c + d*x)**2*csc(c + d*x)**2, x))
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2 \, a^{2} \cot \left (d x + c\right )^{3} + 6 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{2} - 3 \, a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{6 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/6*(2*a^2*cot(d*x + c)^3 + 6*(d*x + c + 1/tan(d*x + c))*a^2 - 3*a^2*(2*c os(d*x + c)/(cos(d*x + c)^2 - 1) + log(cos(d*x + c) + 1) - log(cos(d*x + c ) - 1)))/d
Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, {\left (d x + c\right )} a^{2} - 24 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/24*(a^2*tan(1/2*d*x + 1/2*c)^3 + 6*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*(d*x + c)*a^2 - 24*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 9*a^2*tan(1/2*d*x + 1/2 *c) + (44*a^2*tan(1/2*d*x + 1/2*c)^3 - 9*a^2*tan(1/2*d*x + 1/2*c)^2 - 6*a^ 2*tan(1/2*d*x + 1/2*c) - a^2)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 17.82 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.64 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \] Input:
int((cot(c + d*x)^2*(a + a*sin(c + d*x))^2)/sin(c + d*x)^2,x)
Output:
(a^2*tan(c/2 + (d*x)/2)^2)/(4*d) - (a^2*cot(c/2 + (d*x)/2)^3)/(24*d) - (a^ 2*cot(c/2 + (d*x)/2)^2)/(4*d) + (a^2*tan(c/2 + (d*x)/2)^3)/(24*d) - (2*a^2 *atan((cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2) - sin( c/2 + (d*x)/2))))/d - (a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (3*a^2*cot(c/2 + (d*x)/2))/(8*d) + (3*a^2*tan(c/2 + (d*x)/2))/(8*d)
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-3 \sin \left (d x +c \right )^{3} d x \right )}{3 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 2*cos(c + d*x)*sin(c + d*x)**2 - 3*cos(c + d*x)*sin(c + d*x) - c os(c + d*x) - 3*log(tan((c + d*x)/2))*sin(c + d*x)**3 - 3*sin(c + d*x)**3* d*x))/(3*sin(c + d*x)**3*d)