\(\int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [693]

Optimal result

Integrand size = 29, antiderivative size = 91 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cot ^6(c+d x)}{6 a d}+\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc ^5(c+d x)}{5 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]

[Out]

1/6*cot(d*x+c)^6/a/d+1/8*cot(d*x+c)^8/a/d-1/5*csc(d*x+c)^5/a/d+2/7*csc(d*x+c)^7/a/d-1/9*csc(d*x+c)^9/a/d

Rubi [A] (verified)

Time = 0.13 (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, 2686, 276, 2687, 14} \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cot ^8(c+d x)}{8 a d}+\frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^9(c+d x)}{9 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^5(c+d x)}{5 a d} \]

[In]

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

[Out]

Cot[c + d*x]^6/(6*a*d) + Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]^5/(5*a*d) + (2*Csc[c + d*x]^7)/(7*a*d) - Csc[c
+ d*x]^9/(9*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 ^4(c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^5(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^4 \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^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (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 ^5(c+d x)}{5 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^9(c+d x)}{9 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^4(c+d x) \left (630-504 \csc (c+d x)-840 \csc ^2(c+d x)+720 \csc ^3(c+d x)+315 \csc ^4(c+d x)-280 \csc ^5(c+d x)\right )}{2520 a d} \]

[In]

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

[Out]

(Csc[c + d*x]^4*(630 - 504*Csc[c + d*x] - 840*Csc[c + d*x]^2 + 720*Csc[c + d*x]^3 + 315*Csc[c + d*x]^4 - 280*C
sc[c + d*x]^5))/(2520*a*d)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}}{d a}\)	\(70\)
default	\(-\frac {\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}}{d a}\)	\(70\)
parallelrisch	\(-\frac {\left (3571712+3538944 \cos \left (2 d x +2 c \right )+7665 \sin \left (9 d x +9 c \right )-68985 \sin \left (7 d x +7 c \right )-1014300 \sin \left (5 d x +5 c \right )-1614690 \sin \left (d x +c \right )-1073940 \sin \left (3 d x +3 c \right )+2064384 \cos \left (4 d x +4 c \right )\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42278584320 a d}\)	\(107\)
risch	\(\frac {-\frac {32 i {\mathrm e}^{13 i \left (d x +c \right )}}{5}+4 \,{\mathrm e}^{14 i \left (d x +c \right )}-\frac {384 i {\mathrm e}^{11 i \left (d x +c \right )}}{35}+\frac {4 \,{\mathrm e}^{12 i \left (d x +c \right )}}{3}-\frac {6976 i {\mathrm e}^{9 i \left (d x +c \right )}}{315}+8 \,{\mathrm e}^{10 i \left (d x +c \right )}-\frac {384 i {\mathrm e}^{7 i \left (d x +c \right )}}{35}-8 \,{\mathrm e}^{8 i \left (d x +c \right )}-\frac {32 i {\mathrm e}^{5 i \left (d x +c \right )}}{5}-\frac {4 \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-4 \,{\mathrm e}^{4 i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\)	\(149\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d/a*(1/9*csc(d*x+c)^9-1/8*csc(d*x+c)^8-2/7*csc(d*x+c)^7+1/3*csc(d*x+c)^6+1/5*csc(d*x+c)^5-1/4*csc(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {504 \, \cos \left (d x + c\right )^{4} - 288 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 64}{2520 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(504*cos(d*x + c)^4 - 288*cos(d*x + c)^2 - 105*(6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*sin(d*x + c)
+ 64)/((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)*sin(d*x
+ c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720*sin(d*x + c)^2 + 315*sin(d*x + c) -
280)/(a*d*sin(d*x + c)^9)

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720*sin(d*x + c)^2 + 315*sin(d*x + c) -
280)/(a*d*sin(d*x + c)^9)

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{4}-\frac {{\sin \left (c+d\,x\right )}^4}{5}-\frac {{\sin \left (c+d\,x\right )}^3}{3}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {\sin \left (c+d\,x\right )}{8}-\frac {1}{9}}{a\,d\,{\sin \left (c+d\,x\right )}^9} \]

[In]

int(cos(c + d*x)^7/(sin(c + d*x)^10*(a + a*sin(c + d*x))),x)

[Out]

(sin(c + d*x)/8 + (2*sin(c + d*x)^2)/7 - sin(c + d*x)^3/3 - sin(c + d*x)^4/5 + sin(c + d*x)^5/4 - 1/9)/(a*d*si
n(c + d*x)^9)