Integrand size = 21, antiderivative size = 66 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d} \]
arctanh(cos(d*x+c))/a^2/d-2*cot(d*x+c)/a^2/d-1/3*cot(d*x+c)^3/a^2/d+cot(d* x+c)*csc(d*x+c)/a^2/d
Time = 1.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.83 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^4 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (-9 \cos (c+d x)+5 \cos (3 (c+d x))+6 \left (2 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^3(c+d x)+\sin (2 (c+d x))\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{96 a^2 d (1+\sin (c+d x))^2} \]
((1 + Cot[(c + d*x)/2])^4*Sec[(c + d*x)/2]^2*(-9*Cos[c + d*x] + 5*Cos[3*(c + d*x)] + 6*(2*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d* x]^3 + Sin[2*(c + d*x)]))*Tan[(c + d*x)/2])/(96*a^2*d*(1 + Sin[c + d*x])^2 )
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3187, 3042, 3236, 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.
\(\displaystyle \int \frac {\cot ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a \sin (c+d x)+a)^2}dx\) |
\(\Big \downarrow \) 3187 |
\(\displaystyle \frac {\int \csc ^4(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^2}{\sin (c+d x)^4}dx}{a^4}\) |
\(\Big \downarrow \) 3236 |
\(\displaystyle \frac {\int \left (a^2 \csc ^4(c+d x)-2 a^2 \csc ^3(c+d x)+a^2 \csc ^2(c+d x)\right )dx}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}}{a^4}\) |
((a^2*ArcTanh[Cos[c + d*x]])/d - (2*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)/a^4
3.5.28.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _), x_Symbol] :> Simp[a^p Int[Sin[e + f*x]^p/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ [p, 2*m]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-21 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{2}}\) | \(94\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {7}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{8 d \,a^{2}}\) | \(98\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {7}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{8 d \,a^{2}}\) | \(98\) |
risch | \(-\frac {2 \left (3 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-12 i {\mathrm e}^{2 i \left (d x +c \right )}+5 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}\) | \(111\) |
norman | \(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {29 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {17 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(207\) |
1/24*(tan(1/2*d*x+1/2*c)^3-cot(1/2*d*x+1/2*c)^3-6*tan(1/2*d*x+1/2*c)^2+6*c ot(1/2*d*x+1/2*c)^2-24*ln(tan(1/2*d*x+1/2*c))+21*tan(1/2*d*x+1/2*c)-21*cot (1/2*d*x+1/2*c))/d/a^2
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {10 \, \cos \left (d x + c\right )^{3} - 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]
-1/6*(10*cos(d*x + c)^3 - 3*(cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/ 2)*sin(d*x + c) + 3*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin( d*x + c) + 6*cos(d*x + c)*sin(d*x + c) - 12*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (64) = 128\).
Time = 0.22 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.32 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{2} \sin \left (d x + c\right )^{3}}}{24 \, d} \]
1/24*((21*sin(d*x + c)/(cos(d*x + c) + 1) - 6*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 24*log(sin(d*x + c)/( cos(d*x + c) + 1))/a^2 + (6*sin(d*x + c)/(cos(d*x + c) + 1) - 21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)*(cos(d*x + c) + 1)^3/(a^2*sin(d*x + c)^3)) /d
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.94 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {24 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {44 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 21 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{24 \, d} \]
-1/24*(24*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (44*tan(1/2*d*x + 1/2*c)^3 - 21*tan(1/2*d*x + 1/2*c)^2 + 6*tan(1/2*d*x + 1/2*c) - 1)/(a^2*tan(1/2*d*x + 1/2*c)^3) - (a^4*tan(1/2*d*x + 1/2*c)^3 - 6*a^4*tan(1/2*d*x + 1/2*c)^2 + 21*a^4*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 10.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.80 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{3}\right )}{8\,a^2\,d} \]