\(\int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [734]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d} \]

[Out]

1/8*arctanh(cos(d*x+c))/a^2/d-2/5*cot(d*x+c)^5/a^2/d-1/7*cot(d*x+c)^7/a^2/d+1/8*cot(d*x+c)*csc(d*x+c)/a^2/d-7/
12*cot(d*x+c)*csc(d*x+c)^3/a^2/d+1/3*cot(d*x+c)*csc(d*x+c)^5/a^2/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2788, 3852, 8, 3853, 3855} \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d} \]

[In]

Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]

[Out]

ArcTanh[Cos[c + d*x]]/(8*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a^2*d) + (Cot[c + d*x]*Csc[
c + d*x])/(8*a^2*d) - (7*Cot[c + d*x]*Csc[c + d*x]^3)/(12*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(3*a^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2788

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[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])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (a^6 \csc ^2(c+d x)-2 a^6 \csc ^3(c+d x)-a^6 \csc ^4(c+d x)+4 a^6 \csc ^5(c+d x)-a^6 \csc ^6(c+d x)-2 a^6 \csc ^7(c+d x)+a^6 \csc ^8(c+d x)\right ) \, dx}{a^8} \\ & = \frac {\int \csc ^2(c+d x) \, dx}{a^2}-\frac {\int \csc ^4(c+d x) \, dx}{a^2}-\frac {\int \csc ^6(c+d x) \, dx}{a^2}+\frac {\int \csc ^8(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^7(c+d x) \, dx}{a^2}+\frac {4 \int \csc ^5(c+d x) \, dx}{a^2} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {5 \int \csc ^5(c+d x) \, dx}{3 a^2}+\frac {3 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}+\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d} \\ & = \frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {5 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac {3 \int \csc (c+d x) \, dx}{2 a^2} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {5 \int \csc (c+d x) \, dx}{8 a^2} \\ & = \frac {\text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(251\) vs. \(2(124)=248\).

Time = 1.99 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.02 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^7(c+d x) \left (5880 \cos (c+d x)+2184 \cos (3 (c+d x))-168 \cos (5 (c+d x))-216 \cos (7 (c+d x))-3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-2170 \sin (2 (c+d x))+2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-3080 \sin (4 (c+d x))-735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-210 \sin (6 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{53760 a^2 d} \]

[In]

Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/53760*(Csc[c + d*x]^7*(5880*Cos[c + d*x] + 2184*Cos[3*(c + d*x)] - 168*Cos[5*(c + d*x)] - 216*Cos[7*(c + d*
x)] - 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 2170*Sin[2*(c + d*x)
] + 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 3080*Sin[4*(c
+ d*x)] - 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*
(c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(a^2*d)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.61

method	result	size
parallelrisch	\(\frac {15 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-70 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+70 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+210 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-210 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-525 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+525 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+210 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-210 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1155 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{13440 d \,a^{2}}\)	\(200\)
derivativedivides	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{128 d \,a^{2}}\)	\(202\)
default	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{128 d \,a^{2}}\)	\(202\)
risch	\(-\frac {840 i {\mathrm e}^{12 i \left (d x +c \right )}+105 \,{\mathrm e}^{13 i \left (d x +c \right )}-3360 i {\mathrm e}^{10 i \left (d x +c \right )}+1540 \,{\mathrm e}^{11 i \left (d x +c \right )}+840 i {\mathrm e}^{8 i \left (d x +c \right )}+1085 \,{\mathrm e}^{9 i \left (d x +c \right )}-6720 i {\mathrm e}^{6 i \left (d x +c \right )}+1176 i {\mathrm e}^{4 i \left (d x +c \right )}-1085 \,{\mathrm e}^{5 i \left (d x +c \right )}-672 i {\mathrm e}^{2 i \left (d x +c \right )}-1540 \,{\mathrm e}^{3 i \left (d x +c \right )}+216 i-105 \,{\mathrm e}^{i \left (d x +c \right )}}{420 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}\)	\(204\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/13440*(15*tan(1/2*d*x+1/2*c)^7-15*cot(1/2*d*x+1/2*c)^7-70*tan(1/2*d*x+1/2*c)^6+70*cot(1/2*d*x+1/2*c)^6+63*ta
n(1/2*d*x+1/2*c)^5-63*cot(1/2*d*x+1/2*c)^5+210*tan(1/2*d*x+1/2*c)^4-210*cot(1/2*d*x+1/2*c)^4-525*tan(1/2*d*x+1
/2*c)^3+525*cot(1/2*d*x+1/2*c)^3+210*tan(1/2*d*x+1/2*c)^2-210*cot(1/2*d*x+1/2*c)^2-1680*ln(tan(1/2*d*x+1/2*c))
+1155*tan(1/2*d*x+1/2*c)-1155*cot(1/2*d*x+1/2*c))/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {432 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1680*(432*cos(d*x + c)^7 - 672*cos(d*x + c)^5 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 -
1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*
log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 70*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x
+ c))/((a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**8/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (112) = 224\).

Time = 0.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.53 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {210 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {525 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {210 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {70 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2}} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {525 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1155 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/13440*((1155*sin(d*x + c)/(cos(d*x + c) + 1) + 210*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 525*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 + 210*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 70*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^2 - 1680*log(sin(d*x + c)/(cos(
d*x + c) + 1))/a^2 + (70*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 210*sin(d*
x + c)^3/(cos(d*x + c) + 1)^3 + 525*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 210*sin(d*x + c)^5/(cos(d*x + c) + 1
)^5 - 1155*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 15)*(cos(d*x + c) + 1)^7/(a^2*sin(d*x + c)^7))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (112) = 224\).

Time = 0.36 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.98 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {4356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {15 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 525 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1155 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{14}}}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/13440*(1680*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (4356*tan(1/2*d*x + 1/2*c)^7 - 1155*tan(1/2*d*x + 1/2*c)^6
- 210*tan(1/2*d*x + 1/2*c)^5 + 525*tan(1/2*d*x + 1/2*c)^4 - 210*tan(1/2*d*x + 1/2*c)^3 - 63*tan(1/2*d*x + 1/2
*c)^2 + 70*tan(1/2*d*x + 1/2*c) - 15)/(a^2*tan(1/2*d*x + 1/2*c)^7) - (15*a^12*tan(1/2*d*x + 1/2*c)^7 - 70*a^12
*tan(1/2*d*x + 1/2*c)^6 + 63*a^12*tan(1/2*d*x + 1/2*c)^5 + 210*a^12*tan(1/2*d*x + 1/2*c)^4 - 525*a^12*tan(1/2*
d*x + 1/2*c)^3 + 210*a^12*tan(1/2*d*x + 1/2*c)^2 + 1155*a^12*tan(1/2*d*x + 1/2*c))/a^14)/d

Mupad [B] (verification not implemented)

Time = 11.00 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.12 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+70\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{13440\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^8*(a + a*sin(c + d*x))^2),x)

[Out]

-(15*cos(c/2 + (d*x)/2)^14 - 15*sin(c/2 + (d*x)/2)^14 + 70*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 - 70*cos(c
/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) - 63*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 210*cos(c/2 + (d*x)/2)^3
*sin(c/2 + (d*x)/2)^11 + 525*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 210*cos(c/2 + (d*x)/2)^5*sin(c/2 + (
d*x)/2)^9 - 1155*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 1155*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 +
210*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 525*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 210*cos(c/2 +
(d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 63*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 1680*log(sin(c/2 + (d*x)/2
)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)/(13440*a^2*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (
d*x)/2)^7)