Integrand size = 21, antiderivative size = 96 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {11 \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))} \] Output:
11/2*arctanh(cos(d*x+c))/a^3/d-5*cot(d*x+c)/a^3/d-1/3*cot(d*x+c)^3/a^3/d+3 /2*cot(d*x+c)*csc(d*x+c)/a^3/d-4*cot(d*x+c)/a^3/d/(1+csc(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(251\) vs. \(2(96)=192\).
Time = 4.57 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.61 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^5 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (-4 \sin ^8\left (\frac {1}{2} (c+d x)\right )-8 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x) (-2+7 \sin (c+d x))+\frac {1}{4} \sin ^4(c+d x) \left (-8+\cot \left (\frac {1}{2} (c+d x)\right )+28 \sin (c+d x)\right )-\frac {1}{2} \sin ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^3(c+d x) \left (9+\left (-28+66 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-66 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )+\sin ^4\left (\frac {1}{2} (c+d x)\right ) \sin ^2(c+d x) \left (9-2 \left (62+33 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-33 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )\right )}{12 a^3 d (1+\sin (c+d x))^3} \] Input:
Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]
Output:
-1/12*((1 + Cot[(c + d*x)/2])^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2*(-4*Sin[ (c + d*x)/2]^8 - 8*Sin[(c + d*x)/2]^6*Sin[c + d*x]*(-2 + 7*Sin[c + d*x]) + (Sin[c + d*x]^4*(-8 + Cot[(c + d*x)/2] + 28*Sin[c + d*x]))/4 - (Sin[(c + d*x)/2]^2*Sin[c + d*x]^3*(9 + (-28 + 66*Log[Cos[(c + d*x)/2]] - 66*Log[Sin [(c + d*x)/2]])*Sin[c + d*x]))/2 + Sin[(c + d*x)/2]^4*Sin[c + d*x]^2*(9 - 2*(62 + 33*Log[Cos[(c + d*x)/2]] - 33*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]) ))/(a^3*d*(1 + Sin[c + d*x])^3)
Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a \csc ^4(c+d x)-3 a \csc ^3(c+d x)+4 a \csc ^2(c+d x)-4 a \csc (c+d x)+4 a-\frac {4 a}{\csc (c+d x)+1}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {11 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {5 a \cot (c+d x)}{d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {4 a \cot (c+d x)}{d (\csc (c+d x)+1)}}{a^4}\) |
Input:
Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]
Output:
((11*a*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*Cot[c + d*x])/d - (a*Cot[c + d* x]^3)/(3*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (4*a*Cot[c + d*x])/( d*(1 + Csc[c + d*x])))/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 = 4.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {64}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {19}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}\) | \(113\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {64}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {19}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}\) | \(113\) |
| risch | \(-\frac {33 i {\mathrm e}^{5 i \left (d x +c \right )}-96 \,{\mathrm e}^{4 i \left (d x +c \right )}+33 \,{\mathrm e}^{6 i \left (d x +c \right )}-60 i {\mathrm e}^{3 i \left (d x +c \right )}+123 \,{\mathrm e}^{2 i \left (d x +c \right )}+19 i {\mathrm e}^{i \left (d x +c \right )}-52}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{3} d}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}\) | \(148\) |
Input:
int(cot(d*x+c)^4/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/8/d/a^3*(1/3*tan(1/2*d*x+1/2*c)^3-3*tan(1/2*d*x+1/2*c)^2+19*tan(1/2*d*x+ 1/2*c)-64/(tan(1/2*d*x+1/2*c)+1)-1/3/tan(1/2*d*x+1/2*c)^3+3/tan(1/2*d*x+1/ 2*c)^2-19/tan(1/2*d*x+1/2*c)-44*ln(tan(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (90) = 180\).
Time = 0.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.15 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {104 \, \cos \left (d x + c\right )^{4} + 38 \, \cos \left (d x + c\right )^{3} - 156 \, \cos \left (d x + c\right )^{2} + 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (52 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )^{2} - 45 \, \cos \left (d x + c\right ) - 24\right )} \sin \left (d x + c\right ) - 42 \, \cos \left (d x + c\right ) + 48}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d - {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/12*(104*cos(d*x + c)^4 + 38*cos(d*x + c)^3 - 156*cos(d*x + c)^2 + 33*(co s(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d *x + c) - 1)*sin(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) - 33*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sin(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 2*(52*cos(d*x + c)^ 3 + 33*cos(d*x + c)^2 - 45*cos(d*x + c) - 24)*sin(d*x + c) - 42*cos(d*x + c) + 48)/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d - (a^3*d*c os(d*x + c)^3 + a^3*d*cos(d*x + c)^2 - a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c))
\[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)**4/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**4/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (90) = 180\).
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.07 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {249 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1}{\frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {57 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}} - \frac {132 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
1/24*((8*sin(d*x + c)/(cos(d*x + c) + 1) - 48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 249*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)/(a^3*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (57*sin (d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + sin (d*x + c)^3/(cos(d*x + c) + 1)^3)/a^3 - 132*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {132 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {192}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {242 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 57 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/24*(132*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + 192/(a^3*(tan(1/2*d*x + 1/ 2*c) + 1)) - (242*tan(1/2*d*x + 1/2*c)^3 - 57*tan(1/2*d*x + 1/2*c)^2 + 9*t an(1/2*d*x + 1/2*c) - 1)/(a^3*tan(1/2*d*x + 1/2*c)^3) - (a^6*tan(1/2*d*x + 1/2*c)^3 - 9*a^6*tan(1/2*d*x + 1/2*c)^2 + 57*a^6*tan(1/2*d*x + 1/2*c))/a^ 9)/d
Time = 17.58 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^3\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}-\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d} \] Input:
int(cot(c + d*x)^4/(a + a*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^3/(24*a^3*d) - (3*tan(c/2 + (d*x)/2)^2)/(8*a^3*d) - (11 *log(tan(c/2 + (d*x)/2)))/(2*a^3*d) - (16*tan(c/2 + (d*x)/2)^2 - (8*tan(c/ 2 + (d*x)/2))/3 + 83*tan(c/2 + (d*x)/2)^3 + 1/3)/(d*(8*a^3*tan(c/2 + (d*x) /2)^3 + 8*a^3*tan(c/2 + (d*x)/2)^4)) + (19*tan(c/2 + (d*x)/2))/(8*a^3*d)
Time = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-132 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-132 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+306 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}{24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \] Input:
int(cot(d*x+c)^4/(a+a*sin(d*x+c))^3,x)
Output:
( - 132*log(tan((c + d*x)/2))*tan((c + d*x)/2)**4 - 132*log(tan((c + d*x)/ 2))*tan((c + d*x)/2)**3 + tan((c + d*x)/2)**7 - 8*tan((c + d*x)/2)**6 + 48 *tan((c + d*x)/2)**5 + 306*tan((c + d*x)/2)**4 - 48*tan((c + d*x)/2)**2 + 8*tan((c + d*x)/2) - 1)/(24*tan((c + d*x)/2)**3*a**3*d*(tan((c + d*x)/2) + 1))