\(\int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [716]

Optimal result

Integrand size = 27, antiderivative size = 142 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}+\frac {5 \text {arctanh}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d} \]

[Out]

x/a+5/16*arctanh(cos(d*x+c))/a/d+cot(d*x+c)/a/d-1/3*cot(d*x+c)^3/a/d+1/5*cot(d*x+c)^5/a/d-5/16*cot(d*x+c)*csc(
d*x+c)/a/d+5/24*cot(d*x+c)^3*csc(d*x+c)/a/d-1/6*cot(d*x+c)^5*csc(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2918, 2691, 3855, 3554, 8} \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{16 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {x}{a} \]

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

x/a + (5*ArcTanh[Cos[c + d*x]])/(16*a*d) + Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d
) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) - (Cot[c + d*x]^5*Csc[c
+ d*x])/(6*a*d)

Rule 8

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

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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 \cot ^6(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc (c+d x) \, dx}{a} \\ & = \frac {\cot ^5(c+d x)}{5 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac {5 \int \cot ^4(c+d x) \csc (c+d x) \, dx}{6 a}+\frac {\int \cot ^4(c+d x) \, dx}{a} \\ & = -\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{8 a}-\frac {\int \cot ^2(c+d x) \, dx}{a} \\ & = \frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac {5 \int \csc (c+d x) \, dx}{16 a}+\frac {\int 1 \, dx}{a} \\ & = \frac {x}{a}+\frac {5 \text {arctanh}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(317\) vs. \(2(142)=284\).

Time = 1.71 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.23 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^6(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-2400 c-2400 d x+900 \cos (c+d x)+50 \cos (3 (c+d x))-1440 c \cos (4 (c+d x))-1440 d x \cos (4 (c+d x))+330 \cos (5 (c+d x))+240 c \cos (6 (c+d x))+240 d x \cos (6 (c+d x))-750 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-450 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+75 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \left (16 (c+d x)+5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+750 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+450 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-75 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1200 \sin (2 (c+d x))+768 \sin (4 (c+d x))-368 \sin (6 (c+d x))\right )}{7680 a d (1+\sin (c+d x))} \]

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

-1/7680*(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2400*c - 2400*d*x + 900*Cos[c + d*x] + 50*Co
s[3*(c + d*x)] - 1440*c*Cos[4*(c + d*x)] - 1440*d*x*Cos[4*(c + d*x)] + 330*Cos[5*(c + d*x)] + 240*c*Cos[6*(c +
d*x)] + 240*d*x*Cos[6*(c + d*x)] - 750*Log[Cos[(c + d*x)/2]] - 450*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 7
5*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*(16*(c + d*x) + 5*Log[Cos[(c + d*x)/2]] - 5*Lo
g[Sin[(c + d*x)/2]]) + 750*Log[Sin[(c + d*x)/2]] + 450*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 75*Cos[6*(c +
d*x)]*Log[Sin[(c + d*x)/2]] - 1200*Sin[2*(c + d*x)] + 768*Sin[4*(c + d*x)] - 368*Sin[6*(c + d*x)]))/(a*d*(1 +
Sin[c + d*x]))

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.25

method	result	size
parallelrisch	\(\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+225 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-225 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1920 d x -600 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1320 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1320 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d a}\)	\(178\)
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {15}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {44}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}\)	\(188\)
default	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {15}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {44}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}\)	\(188\)
risch	\(\frac {x}{a}+\frac {720 i {\mathrm e}^{10 i \left (d x +c \right )}+165 \,{\mathrm e}^{11 i \left (d x +c \right )}-2160 i {\mathrm e}^{8 i \left (d x +c \right )}+25 \,{\mathrm e}^{9 i \left (d x +c \right )}+3680 i {\mathrm e}^{6 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}-3360 i {\mathrm e}^{4 i \left (d x +c \right )}+450 \,{\mathrm e}^{5 i \left (d x +c \right )}+1488 i {\mathrm e}^{2 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-368 i+165 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d a}\)	\(197\)
norman	\(\frac {\frac {x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1}{384 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d a}+\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d a}-\frac {77 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}+\frac {25 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {25 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {77 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}+\frac {11 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d a}-\frac {13 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}-\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1920 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {59 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {73 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {191 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}\)	\(397\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/1920*(5*tan(1/2*d*x+1/2*c)^6-5*cot(1/2*d*x+1/2*c)^6-12*tan(1/2*d*x+1/2*c)^5+12*cot(1/2*d*x+1/2*c)^5-45*tan(1
/2*d*x+1/2*c)^4+45*cot(1/2*d*x+1/2*c)^4+140*tan(1/2*d*x+1/2*c)^3-140*cot(1/2*d*x+1/2*c)^3+225*tan(1/2*d*x+1/2*
c)^2-225*cot(1/2*d*x+1/2*c)^2+1920*d*x-600*ln(tan(1/2*d*x+1/2*c))-1320*tan(1/2*d*x+1/2*c)+1320*cot(1/2*d*x+1/2
*c))/d/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {480 \, d x \cos \left (d x + c\right )^{6} - 1440 \, d x \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{5} + 1440 \, d x \cos \left (d x + c\right )^{2} - 400 \, \cos \left (d x + c\right )^{3} - 480 \, d x + 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (23 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 150 \, \cos \left (d x + c\right )}{480 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/480*(480*d*x*cos(d*x + c)^6 - 1440*d*x*cos(d*x + c)^4 + 330*cos(d*x + c)^5 + 1440*d*x*cos(d*x + c)^2 - 400*c
os(d*x + c)^3 - 480*d*x + 75*(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) - 75*(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) - 32*(23*co
s(d*x + c)^5 - 35*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) + 150*cos(d*x + c))/(a*d*cos(d*x + c)^6 - 3*a
*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (130) = 260\).

Time = 0.38 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.10 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {1320 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac {3840 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {600 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {225 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1320 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1920*((1320*sin(d*x + c)/(cos(d*x + c) + 1) - 225*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 + 45*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 12*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6)/a - 3840*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 600*log(sin(d*x + c)/(c
os(d*x + c) + 1))/a - (12*sin(d*x + c)/(cos(d*x + c) + 1) + 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d
*x + c)^3/(cos(d*x + c) + 1)^3 - 225*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1320*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 5)*(cos(d*x + c) + 1)^6/(a*sin(d*x + c)^6))/d

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.58 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {1920 \, {\left (d x + c\right )}}{a} - \frac {600 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1920*(1920*(d*x + c)/a - 600*log(abs(tan(1/2*d*x + 1/2*c)))/a + (5*a^5*tan(1/2*d*x + 1/2*c)^6 - 12*a^5*tan(1
/2*d*x + 1/2*c)^5 - 45*a^5*tan(1/2*d*x + 1/2*c)^4 + 140*a^5*tan(1/2*d*x + 1/2*c)^3 + 225*a^5*tan(1/2*d*x + 1/2
*c)^2 - 1320*a^5*tan(1/2*d*x + 1/2*c))/a^6 + (1470*tan(1/2*d*x + 1/2*c)^6 + 1320*tan(1/2*d*x + 1/2*c)^5 - 225*
tan(1/2*d*x + 1/2*c)^4 - 140*tan(1/2*d*x + 1/2*c)^3 + 45*tan(1/2*d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2*c) - 5)
/(a*tan(1/2*d*x + 1/2*c)^6))/d

Mupad [B] (verification not implemented)

Time = 12.09 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.91 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+12\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3840\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+600\,\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 )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^7*(a + a*sin(c + d*x))),x)

[Out]

-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 + 12*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 - 12*cos(c/2
+ (d*x)/2)^11*sin(c/2 + (d*x)/2) + 45*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 140*cos(c/2 + (d*x)/2)^3*s
in(c/2 + (d*x)/2)^9 - 225*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 1320*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x
)/2)^7 - 1320*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 + 225*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 140*
cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 - 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 3840*atan((16*cos(
c/2 + (d*x)/2) - 5*sin(c/2 + (d*x)/2))/(5*cos(c/2 + (d*x)/2) + 16*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^6*si
n(c/2 + (d*x)/2)^6 + 600*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6)
/(1920*a*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6)