\(\int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [692]

Optimal result

Integrand size = 29, antiderivative size = 91 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cot ^6(c+d x)}{6 a d}-\frac {\cot ^8(c+d x)}{8 a d}+\frac {\csc ^3(c+d x)}{3 a d}-\frac {2 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^7(c+d x)}{7 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2914, 2687, 14, 2686, 276} \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cot ^8(c+d x)}{8 a d}-\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {2 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{3 a d} \]

[In]

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

[Out]

-1/6*Cot[c + d*x]^6/(a*d) - Cot[c + d*x]^8/(8*a*d) + Csc[c + d*x]^3/(3*a*d) - (2*Csc[c + d*x]^5)/(5*a*d) + Csc
[c + d*x]^7/(7*a*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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^3(c+d x) \left (280-210 \csc (c+d x)-336 \csc ^2(c+d x)+280 \csc ^3(c+d x)+120 \csc ^4(c+d x)-105 \csc ^5(c+d x)\right )}{840 a d} \]

[In]

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

[Out]

(Csc[c + d*x]^3*(280 - 210*Csc[c + d*x] - 336*Csc[c + d*x]^2 + 280*Csc[c + d*x]^3 + 120*Csc[c + d*x]^4 - 105*C
sc[c + d*x]^5))/(840*a*d)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d a}\)	\(70\)
default	\(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d a}\)	\(70\)
parallelrisch	\(-\frac {\left (627375+716520 \cos \left (2 d x +2 c \right )-2555 \cos \left (8 d x +8 c \right )-286720 \sin \left (5 d x +5 c \right )+20440 \cos \left (6 d x +6 c \right )-704512 \sin \left (d x +c \right )+57344 \sin \left (3 d x +3 c \right )+358540 \cos \left (4 d x +4 c \right )\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3523215360 a d}\)	\(107\)
risch	\(-\frac {4 i \left (-105 i {\mathrm e}^{12 i \left (d x +c \right )}+70 \,{\mathrm e}^{13 i \left (d x +c \right )}-140 i {\mathrm e}^{10 i \left (d x +c \right )}-14 \,{\mathrm e}^{11 i \left (d x +c \right )}-350 i {\mathrm e}^{8 i \left (d x +c \right )}+172 \,{\mathrm e}^{9 i \left (d x +c \right )}-140 i {\mathrm e}^{6 i \left (d x +c \right )}-172 \,{\mathrm e}^{7 i \left (d x +c \right )}-105 i {\mathrm e}^{4 i \left (d x +c \right )}+14 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\)	\(150\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^9/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{4} - 140 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 35}{840 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/840*(210*cos(d*x + c)^4 - 140*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^4 - 28*cos(d*x + c)^2 + 8)*sin(d*x + c) +
35)/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/840*(280*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 336*sin(d*x + c)^3 + 280*sin(d*x + c)^2 + 120*sin(d*x + c) -
105)/(a*d*sin(d*x + c)^8)

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/840*(280*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 336*sin(d*x + c)^3 + 280*sin(d*x + c)^2 + 120*sin(d*x + c) -
105)/(a*d*sin(d*x + c)^8)

Mupad [B] (verification not implemented)

Time = 9.89 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {280\,{\sin \left (c+d\,x\right )}^5-210\,{\sin \left (c+d\,x\right )}^4-336\,{\sin \left (c+d\,x\right )}^3+280\,{\sin \left (c+d\,x\right )}^2+120\,\sin \left (c+d\,x\right )-105}{840\,a\,d\,{\sin \left (c+d\,x\right )}^8} \]

[In]

int(cos(c + d*x)^7/(sin(c + d*x)^9*(a + a*sin(c + d*x))),x)

[Out]

(120*sin(c + d*x) + 280*sin(c + d*x)^2 - 336*sin(c + d*x)^3 - 210*sin(c + d*x)^4 + 280*sin(c + d*x)^5 - 105)/(
840*a*d*sin(c + d*x)^8)