\(\int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [748]

Optimal result

Integrand size = 27, antiderivative size = 124 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d} \]

[Out]

7/16*arctanh(cos(d*x+c))/a^3/d+4/3*cot(d*x+c)^3/a^3/d+3/5*cot(d*x+c)^5/a^3/d+7/16*cot(d*x+c)*csc(d*x+c)/a^3/d-
17/24*cot(d*x+c)*csc(d*x+c)^3/a^3/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a^3/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2954, 2952, 2687, 30, 2691, 3853, 3855, 14} \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]

[Out]

(7*ArcTanh[Cos[c + d*x]])/(16*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (3*Cot[c + d*x]^5)/(5*a^3*d) + (7*Cot[c
+ d*x]*Csc[c + d*x])/(16*a^3*d) - (17*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/
(6*a^3*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 ^2(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^2(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+a^3 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3} \\ & = -\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc ^5(c+d x) \, dx}{6 a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac {\text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc ^3(c+d x) \, dx}{8 a^3}-\frac {3 \int \csc (c+d x) \, dx}{8 a^3}-\frac {3 \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc (c+d x) \, dx}{16 a^3} \\ & = \frac {7 \text {arctanh}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (-704 \cot \left (\frac {1}{2} (c+d x)\right )+210 \csc ^2\left (\frac {1}{2} (c+d x)\right )+840 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-840 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-210 \sec ^2\left (\frac {1}{2} (c+d x)\right )+90 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )-544 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (-5+18 \sin (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (-90+34 \sin (c+d x))+704 \tan \left (\frac {1}{2} (c+d x)\right )-36 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{1920 a^3 d (1+\sin (c+d x))^3} \]

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-704*Cot[(c + d*x)/2] + 210*Csc[(c + d*x)/2]^2 + 840*Log[Cos[(c + d*
x)/2]] - 840*Log[Sin[(c + d*x)/2]] - 210*Sec[(c + d*x)/2]^2 + 90*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 5
44*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6*(-5 + 18*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-90 + 3
4*Sin[c + d*x]) + 704*Tan[(c + d*x)/2] - 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*a^3*d*(1 + Sin[c + d*x
])^3)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40

method	result	size
parallelrisch	\(\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+36 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-105 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+600 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-600 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d \,a^{3}}\)	\(174\)
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{64 d \,a^{3}}\)	\(176\)
default	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{64 d \,a^{3}}\)	\(176\)
risch	\(-\frac {240 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{11 i \left (d x +c \right )}-2160 i {\mathrm e}^{8 i \left (d x +c \right )}+365 \,{\mathrm e}^{9 i \left (d x +c \right )}+1760 i {\mathrm e}^{6 i \left (d x +c \right )}-1110 \,{\mathrm e}^{7 i \left (d x +c \right )}-480 i {\mathrm e}^{4 i \left (d x +c \right )}-1110 \,{\mathrm e}^{5 i \left (d x +c \right )}+816 i {\mathrm e}^{2 i \left (d x +c \right )}+365 \,{\mathrm e}^{3 i \left (d x +c \right )}-176 i+105 \,{\mathrm e}^{i \left (d x +c \right )}}{120 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{3}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{3}}\)	\(192\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,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)^6-36*tan(1/2*d*x+1/2*c)^5+36*cot(1/2*d*x+1/2*c)^5+105*tan(
1/2*d*x+1/2*c)^4-105*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)^3-15*tan(1/2*d*x+1/2
*c)^2+15*cot(1/2*d*x+1/2*c)^2-840*ln(tan(1/2*d*x+1/2*c))+600*tan(1/2*d*x+1/2*c)-600*cot(1/2*d*x+1/2*c))/d/a^3

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.58 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (11 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )}{480 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(210*cos(d*x + c)^5 - 80*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) + 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) - 32*(11*cos(d*x + c)^5 - 20*cos(d*x + c)^3)*sin(d*x + c) - 210*cos(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**7/(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. 274 vs. \(2 (112) = 224\).

Time = 0.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {600 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {600 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{3} \sin \left (d x + c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/1920*((600*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 105*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6)/a^3 - 840*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + (36*sin(d*x + c)/(cos(d*x
+ c) + 1) - 105*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x +
c)^4/(cos(d*x + c) + 1)^4 - 600*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^3*sin(d*x + c
)^6))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2058 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {5 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 600 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/1920*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2058*tan(1/2*d*x + 1/2*c)^6 - 600*tan(1/2*d*x + 1/2*c)^5 +
15*tan(1/2*d*x + 1/2*c)^4 + 140*tan(1/2*d*x + 1/2*c)^3 - 105*tan(1/2*d*x + 1/2*c)^2 + 36*tan(1/2*d*x + 1/2*c)
- 5)/(a^3*tan(1/2*d*x + 1/2*c)^6) - (5*a^15*tan(1/2*d*x + 1/2*c)^6 - 36*a^15*tan(1/2*d*x + 1/2*c)^5 + 105*a^15
*tan(1/2*d*x + 1/2*c)^4 - 140*a^15*tan(1/2*d*x + 1/2*c)^3 - 15*a^15*tan(1/2*d*x + 1/2*c)^2 + 600*a^15*tan(1/2*
d*x + 1/2*c))/a^18)/d

Mupad [B] (verification not implemented)

Time = 10.35 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+36\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^7*(a + a*sin(c + d*x))^3),x)

[Out]

-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 + 36*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 - 36*cos(c/2
+ (d*x)/2)^11*sin(c/2 + (d*x)/2) - 105*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 140*cos(c/2 + (d*x)/2)^3*
sin(c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 600*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)
/2)^7 + 600*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 15*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 + 105*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^3*d*cos(c/2 + (d*x)/2)^6*sin(c/2
+ (d*x)/2)^6)