\(\int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [643]

Optimal result

Integrand size = 27, antiderivative size = 124 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \]

[Out]

3/16*arctanh(cos(d*x+c))/a^2/d+2/3*cot(d*x+c)^3/a^2/d+2/5*cot(d*x+c)^5/a^2/d+3/16*cot(d*x+c)*csc(d*x+c)/a^2/d-
5/24*cot(d*x+c)*csc(d*x+c)^3/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a^2/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2954, 2952, 2691, 3853, 3855, 2687, 14} \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]

[In]

Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]

[Out]

(3*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (3*Cot[c
+ d*x]*Csc[c + d*x])/(16*a^2*d) - (5*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(
6*a^2*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 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))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cot ^2(c+d x) \csc ^3(c+d x)-2 a^2 \cot ^2(c+d x) \csc ^4(c+d x)+a^2 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^2} \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\int \csc ^5(c+d x) \, dx}{6 a^2}-\frac {\int \csc ^3(c+d x) \, dx}{4 a^2}-\frac {2 \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\int \csc (c+d x) \, dx}{8 a^2}-\frac {\int \csc ^3(c+d x) \, dx}{8 a^2}-\frac {2 \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d} \\ & = \frac {\text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\int \csc (c+d x) \, dx}{16 a^2} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^6(c+d x) \left (1500 \cos (c+d x)-130 \cos (3 (c+d x))-90 \cos (5 (c+d x))-450 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+675 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-270 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+450 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-675 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+270 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))\right )}{7680 a^2 d} \]

[In]

Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/7680*(Csc[c + d*x]^6*(1500*Cos[c + d*x] - 130*Cos[3*(c + d*x)] - 90*Cos[5*(c + d*x)] - 450*Log[Cos[(c + d*x
)/2]] + 675*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 270*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 45*Cos[6*(c
+ d*x)]*Log[Cos[(c + d*x)/2]] + 450*Log[Sin[(c + d*x)/2]] - 675*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 270*C
os[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 45*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 960*Sin[2*(c + d*x)] - 384
*Sin[4*(c + d*x)] + 64*Sin[6*(c + d*x)]))/(a^2*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.35

method	result	size
risch	\(-\frac {45 \,{\mathrm e}^{11 i \left (d x +c \right )}-960 i {\mathrm e}^{8 i \left (d x +c \right )}+65 \,{\mathrm e}^{9 i \left (d x +c \right )}+640 i {\mathrm e}^{6 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}-750 \,{\mathrm e}^{5 i \left (d x +c \right )}+384 i {\mathrm e}^{2 i \left (d x +c \right )}+65 \,{\mathrm e}^{3 i \left (d x +c \right )}-64 i+45 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{2}}\)	\(168\)
parallelrisch	\(\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d \,a^{2}}\)	\(174\)
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{2}}\)	\(176\)
default	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{2}}\)	\(176\)
norman	\(\frac {-\frac {1}{384 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{640 d a}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}-\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d a}+\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {3 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {127 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {79 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2}}\)	\(321\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/120*(45*exp(11*I*(d*x+c))-960*I*exp(8*I*(d*x+c))+65*exp(9*I*(d*x+c))+640*I*exp(6*I*(d*x+c))-750*exp(7*I*(d*
x+c))-750*exp(5*I*(d*x+c))+384*I*exp(2*I*(d*x+c))+65*exp(3*I*(d*x+c))-64*I+45*exp(I*(d*x+c)))/a^2/d/(exp(2*I*(
d*x+c))-1)^6-3/16/d/a^2*ln(exp(I*(d*x+c))-1)+3/16/d/a^2*ln(exp(I*(d*x+c))+1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {90 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 90 \, \cos \left (d x + c\right )}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(90*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 45*(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) + 45*(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) - 64*(2*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*sin(d*x + c) - 90*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. \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*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. 274 vs. \(2 (112) = 224\).

Time = 0.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {240 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac {360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {240 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*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) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 40*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 + 45*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 - 360*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (24*sin(d*x + c)/(cos(d*x +
c) + 1) - 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*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 = 0.36 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {882 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {5 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/1920*(360*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (882*tan(1/2*d*x + 1/2*c)^6 - 240*tan(1/2*d*x + 1/2*c)^5 + 1
5*tan(1/2*d*x + 1/2*c)^4 + 40*tan(1/2*d*x + 1/2*c)^3 - 45*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 + 45*a^10*tan
(1/2*d*x + 1/2*c)^4 - 40*a^10*tan(1/2*d*x + 1/2*c)^3 - 15*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 = 11.00 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^7*(a + a*sin(c + d*x))^2),x)

[Out]

-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(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) - 45*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 40*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^9 + 15*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 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 40*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 + 360*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)