\(\int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [749]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{16 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d} \]

[Out]

-5/16*arctanh(cos(d*x+c))/a^3/d-4/3*cot(d*x+c)^3/a^3/d-cot(d*x+c)^5/a^3/d-1/7*cot(d*x+c)^7/a^3/d-5/16*cot(d*x+
c)*csc(d*x+c)/a^3/d+1/8*cot(d*x+c)*csc(d*x+c)^3/a^3/d+1/2*cot(d*x+c)*csc(d*x+c)^5/a^3/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2788, 3853, 3855, 3852} \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{16 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

[In]

Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7
/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x
]*Csc[c + d*x]^5)/(2*a^3*d)

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^5 \csc ^3(c+d x)-3 a^5 \csc ^4(c+d x)+2 a^5 \csc ^5(c+d x)+2 a^5 \csc ^6(c+d x)-3 a^5 \csc ^7(c+d x)+a^5 \csc ^8(c+d x)\right ) \, dx}{a^8} \\ & = \frac {\int \csc ^3(c+d x) \, dx}{a^3}+\frac {\int \csc ^8(c+d x) \, dx}{a^3}+\frac {2 \int \csc ^5(c+d x) \, dx}{a^3}+\frac {2 \int \csc ^6(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^7(c+d x) \, dx}{a^3} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {\int \csc (c+d x) \, dx}{2 a^3}+\frac {3 \int \csc ^3(c+d x) \, dx}{2 a^3}-\frac {5 \int \csc ^5(c+d x) \, dx}{2 a^3}-\frac {\text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {2 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{4 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {3 \int \csc (c+d x) \, dx}{4 a^3}-\frac {15 \int \csc ^3(c+d x) \, dx}{8 a^3} \\ & = -\frac {5 \text {arctanh}(\cos (c+d x))}{4 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac {15 \int \csc (c+d x) \, dx}{16 a^3} \\ & = -\frac {5 \text {arctanh}(\cos (c+d x))}{16 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.79 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^7(c+d x) \left (-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))-3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+4998 \sin (2 (c+d x))+2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+504 \sin (4 (c+d x))-735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-210 \sin (6 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{21504 a^3 d} \]

[In]

Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^7*(-4704*Cos[c + d*x] + 672*Cos[3*(c + d*x)] + 1120*Cos[5*(c + d*x)] - 160*Cos[7*(c + d*x)] - 36
75*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 4998*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)] + 504*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)]))/(21504*a^3*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.37

method	result	size
risch	\(\frac {105 \,{\mathrm e}^{13 i \left (d x +c \right )}-2016 i {\mathrm e}^{10 i \left (d x +c \right )}-252 \,{\mathrm e}^{11 i \left (d x +c \right )}+5152 i {\mathrm e}^{8 i \left (d x +c \right )}-2499 \,{\mathrm e}^{9 i \left (d x +c \right )}-448 i {\mathrm e}^{6 i \left (d x +c \right )}+1344 i {\mathrm e}^{4 i \left (d x +c \right )}+2499 \,{\mathrm e}^{5 i \left (d x +c \right )}-1120 i {\mathrm e}^{2 i \left (d x +c \right )}+252 \,{\mathrm e}^{3 i \left (d x +c \right )}+160 i-105 \,{\mathrm e}^{i \left (d x +c \right )}}{168 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{3}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{3}}\)	\(192\)
derivativedivides	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{128 d \,a^{3}}\)	\(200\)
default	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{128 d \,a^{3}}\)	\(200\)
parallelrisch	\(\frac {-3 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-105 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-91 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+91 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+609 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-609 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2688 d \,a^{3}}\)	\(200\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/168*(105*exp(13*I*(d*x+c))-2016*I*exp(10*I*(d*x+c))-252*exp(11*I*(d*x+c))+5152*I*exp(8*I*(d*x+c))-2499*exp(9
*I*(d*x+c))-448*I*exp(6*I*(d*x+c))+1344*I*exp(4*I*(d*x+c))+2499*exp(5*I*(d*x+c))-1120*I*exp(2*I*(d*x+c))+252*e
xp(3*I*(d*x+c))+160*I-105*exp(I*(d*x+c)))/a^3/d/(exp(2*I*(d*x+c))-1)^7+5/16/d/a^3*ln(exp(I*(d*x+c))-1)-5/16/d/
a^3*ln(exp(I*(d*x+c))+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {320 \, \cos \left (d x + c\right )^{7} - 1120 \, \cos \left (d x + c\right )^{5} + 896 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (5 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/672*(320*cos(d*x + c)^7 - 1120*cos(d*x + c)^5 + 896*cos(d*x + c)^3 - 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) + 42*(5*cos(d*x + c)^5 - 8*cos(d*x + c)^3 - 5*co
s(d*x + c))*sin(d*x + c))/((a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)*si
n(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**8/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (128) = 256\).

Time = 0.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.25 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {609 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {91 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {105 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {609 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{2688 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2688*((609*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 91*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 105*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 21*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^3 - 840*log(sin(d*x + c)/(cos(d*x +
c) + 1))/a^3 - (21*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 105*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 - 91*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6
09*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3)*(cos(d*x + c) + 1)^7/(a^3*sin(d*x + c)^7))/d

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 609 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 91 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {3 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 91 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 609 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{2688 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2178*tan(1/2*d*x + 1/2*c)^7 - 609*tan(1/2*d*x + 1/2*c)^6 + 6
3*tan(1/2*d*x + 1/2*c)^5 + 91*tan(1/2*d*x + 1/2*c)^4 - 105*tan(1/2*d*x + 1/2*c)^3 + 63*tan(1/2*d*x + 1/2*c)^2
- 21*tan(1/2*d*x + 1/2*c) + 3)/(a^3*tan(1/2*d*x + 1/2*c)^7) + (3*a^18*tan(1/2*d*x + 1/2*c)^7 - 21*a^18*tan(1/2
*d*x + 1/2*c)^6 + 63*a^18*tan(1/2*d*x + 1/2*c)^5 - 105*a^18*tan(1/2*d*x + 1/2*c)^4 + 91*a^18*tan(1/2*d*x + 1/2
*c)^3 + 63*a^18*tan(1/2*d*x + 1/2*c)^2 - 609*a^18*tan(1/2*d*x + 1/2*c))/a^21)/d

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.76 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-21\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2688\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^8*(a + a*sin(c + d*x))^3),x)

[Out]

(3*sin(c/2 + (d*x)/2)^14 - 3*cos(c/2 + (d*x)/2)^14 - 21*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 + 21*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 - 105*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^11 + 91*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 63*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/
2)^9 - 609*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 609*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 63*cos(
c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 91*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 105*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 + 840*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)/(2688*a^3*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)