\(\int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [733]

Optimal result

Integrand size = 27, antiderivative size = 132 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d} \]

[Out]

-7/16*arctanh(cos(d*x+c))/a^2/d+2/5*cot(d*x+c)^5/a^2/d+5/16*cot(d*x+c)*csc(d*x+c)/a^2/d-1/4*cot(d*x+c)^3*csc(d
*x+c)/a^2/d+1/8*cot(d*x+c)*csc(d*x+c)^3/a^2/d-1/6*cot(d*x+c)^3*csc(d*x+c)^3/a^2/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2954, 2952, 2691, 3855, 2687, 30, 3853} \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-7*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d
) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^2*d) - (Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(6*a^2*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

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 2952

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

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e +
f*x])^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 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 \cot ^4(c+d x) \csc ^3(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cot ^4(c+d x) \csc (c+d x)-2 a^2 \cot ^4(c+d x) \csc ^2(c+d x)+a^2 \cot ^4(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cot ^4(c+d x) \csc (c+d x) \, dx}{a^2}+\frac {\int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx}{a^2} \\ & = -\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}-\frac {\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{2 a^2}-\frac {3 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{4 a^2}-\frac {2 \text {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{a^2 d} \\ & = \frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\int \csc ^3(c+d x) \, dx}{8 a^2}+\frac {3 \int \csc (c+d x) \, dx}{8 a^2} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\int \csc (c+d x) \, dx}{16 a^2} \\ & = -\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.56 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^6(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (3360 \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^6(c+d x)+60 \cos (c+d x) (-11+32 \sin (c+d x))+6 \cos (5 (c+d x)) (45+32 \sin (c+d x))+10 \cos (3 (c+d x)) (-89+96 \sin (c+d x))\right )}{7680 a^2 d (1+\sin (c+d x))^2} \]

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(3360*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])
*Sin[c + d*x]^6 + 60*Cos[c + d*x]*(-11 + 32*Sin[c + d*x]) + 6*Cos[5*(c + d*x)]*(45 + 32*Sin[c + d*x]) + 10*Cos
[3*(c + d*x)]*(-89 + 96*Sin[c + d*x])))/(7680*a^2*d*(1 + Sin[c + d*x])^2)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.32

method	result	size
parallelrisch	\(\frac {-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+255 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-255 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d \,a^{2}}\)	\(174\)
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {17}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{64 d \,a^{2}}\)	\(176\)
default	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {17}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{64 d \,a^{2}}\)	\(176\)
risch	\(-\frac {-480 i {\mathrm e}^{10 i \left (d x +c \right )}+135 \,{\mathrm e}^{11 i \left (d x +c \right )}+480 i {\mathrm e}^{8 i \left (d x +c \right )}-445 \,{\mathrm e}^{9 i \left (d x +c \right )}-960 i {\mathrm e}^{6 i \left (d x +c \right )}-330 \,{\mathrm e}^{7 i \left (d x +c \right )}+960 i {\mathrm e}^{4 i \left (d x +c \right )}-330 \,{\mathrm e}^{5 i \left (d x +c \right )}-96 i {\mathrm e}^{2 i \left (d x +c \right )}-445 \,{\mathrm e}^{3 i \left (d x +c \right )}+96 i+135 \,{\mathrm e}^{i \left (d x +c \right )}}{120 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{2}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{2}}\)	\(192\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/1920*(-5*cot(1/2*d*x+1/2*c)^6+5*tan(1/2*d*x+1/2*c)^6+24*cot(1/2*d*x+1/2*c)^5-24*tan(1/2*d*x+1/2*c)^5-15*cot(
1/2*d*x+1/2*c)^4+15*tan(1/2*d*x+1/2*c)^4-120*cot(1/2*d*x+1/2*c)^3+120*tan(1/2*d*x+1/2*c)^3+255*cot(1/2*d*x+1/2
*c)^2-255*tan(1/2*d*x+1/2*c)^2+840*ln(tan(1/2*d*x+1/2*c))+240*cot(1/2*d*x+1/2*c)-240*tan(1/2*d*x+1/2*c))/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.39 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {192 \, \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 270 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 210 \, \cos \left (d x + c\right )}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(192*cos(d*x + c)^5*sin(d*x + c) + 270*cos(d*x + c)^5 - 560*cos(d*x + c)^3 + 105*(cos(d*x + c)^6 - 3*co
s(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) - 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) + 210*cos(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**7/(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. 275 vs. \(2 (120) = 240\).

Time = 0.21 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.08 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {240 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {255 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {255 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {240 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1920*((240*sin(d*x + c)/(cos(d*x + c) + 1) + 255*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 120*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 15*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6)/a^2 - 840*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (24*sin(d*x + c)/(cos(d*
x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 120*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 255*sin(d*x +
c)^4/(cos(d*x + c) + 1)^4 + 240*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^2*sin(d*x +
c)^6))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.63 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {2058 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} + \frac {5 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 255 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (2058*tan(1/2*d*x + 1/2*c)^6 - 240*tan(1/2*d*x + 1/2*c)^5 - 2
55*tan(1/2*d*x + 1/2*c)^4 + 120*tan(1/2*d*x + 1/2*c)^3 + 15*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c) +
5)/(a^2*tan(1/2*d*x + 1/2*c)^6) + (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^10*tan(1/2*d*x + 1/2*c)^5 + 15*a^10*t
an(1/2*d*x + 1/2*c)^4 + 120*a^10*tan(1/2*d*x + 1/2*c)^3 - 255*a^10*tan(1/2*d*x + 1/2*c)^2 - 240*a^10*tan(1/2*d
*x + 1/2*c))/a^12)/d

Mupad [B] (verification not implemented)

Time = 10.84 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.57 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-255\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+255\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^7*(a + a*sin(c + d*x))^2),x)

[Out]

(5*sin(c/2 + (d*x)/2)^12 - 5*cos(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 24*cos(c/2
+ (d*x)/2)^11*sin(c/2 + (d*x)/2) + 15*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 120*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^9 - 255*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 240*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/
2)^7 + 240*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 + 255*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 120*cos
(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 - 15*cos(c/2 + (d*x)/2)^10*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)^6*sin(c/2 + (d*x)/2)^6)/(1920*a^2*d*cos(c/2 + (d*x)/2)^6*sin(c/2
+ (d*x)/2)^6)