\(\int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx\) [695]

Optimal result

Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{4 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}-\frac {\csc ^{11}(c+d x)}{11 a d} \]

[Out]

1/6*csc(d*x+c)^6/a/d-1/7*csc(d*x+c)^7/a/d-1/4*csc(d*x+c)^8/a/d+2/9*csc(d*x+c)^9/a/d+1/10*csc(d*x+c)^10/a/d-1/1
1*csc(d*x+c)^11/a/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{4 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d} \]

[In]

Int[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(4*a*d) + (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c
+ d*x]^10/(10*a*d) - Csc[c + d*x]^11/(11*a*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^{12} (a-x)^3 (a+x)^2}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {a^5 \text {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^5 \text {Subst}\left (\int \left (\frac {a^5}{x^{12}}-\frac {a^4}{x^{11}}-\frac {2 a^3}{x^{10}}+\frac {2 a^2}{x^9}+\frac {a}{x^8}-\frac {1}{x^7}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{4 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}-\frac {\csc ^{11}(c+d x)}{11 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^6(c+d x) \left (2310-1980 \csc (c+d x)-3465 \csc ^2(c+d x)+3080 \csc ^3(c+d x)+1386 \csc ^4(c+d x)-1260 \csc ^5(c+d x)\right )}{13860 a d} \]

[In]

Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

(Csc[c + d*x]^6*(2310 - 1980*Csc[c + d*x] - 3465*Csc[c + d*x]^2 + 3080*Csc[c + d*x]^3 + 1386*Csc[c + d*x]^4 -
1260*Csc[c + d*x]^5))/(13860*a*d)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}}{d a}\)	\(70\)
default	\(-\frac {\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}}{d a}\)	\(70\)
parallelrisch	\(-\frac {\left (947200+1126400 \cos \left (2 d x +2 c \right )+2541 \sin \left (9 d x +9 c \right )-12705 \sin \left (7 d x +7 c \right )-257565 \sin \left (5 d x +5 c \right )-366366 \sin \left (d x +c \right )-371910 \sin \left (3 d x +3 c \right )+506880 \cos \left (4 d x +4 c \right )-231 \sin \left (11 d x +11 c \right )\right ) \left (\sec ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{58133053440 a d}\)	\(118\)
risch	\(-\frac {32 \left (-1980 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}-4400 i {\mathrm e}^{13 i \left (d x +c \right )}+1155 \,{\mathrm e}^{14 i \left (d x +c \right )}-7400 i {\mathrm e}^{11 i \left (d x +c \right )}+1848 \,{\mathrm e}^{12 i \left (d x +c \right )}-4400 i {\mathrm e}^{9 i \left (d x +c \right )}-1848 \,{\mathrm e}^{10 i \left (d x +c \right )}-1980 i {\mathrm e}^{7 i \left (d x +c \right )}-1155 \,{\mathrm e}^{8 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3465 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\)	\(149\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^12/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d/a*(1/11*csc(d*x+c)^11-1/10*csc(d*x+c)^10-2/9*csc(d*x+c)^9+1/4*csc(d*x+c)^8+1/7*csc(d*x+c)^7-1/6*csc(d*x+c
)^6)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1980 \, \cos \left (d x + c\right )^{4} - 880 \, \cos \left (d x + c\right )^{2} - 231 \, {\left (10 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 160}{13860 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/13860*(1980*cos(d*x + c)^4 - 880*cos(d*x + c)^2 - 231*(10*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 1)*sin(d*x + c
) + 160)/((a*d*cos(d*x + c)^10 - 5*a*d*cos(d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*
cos(d*x + c)^2 - a*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**12/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/13860*(2310*sin(d*x + c)^5 - 1980*sin(d*x + c)^4 - 3465*sin(d*x + c)^3 + 3080*sin(d*x + c)^2 + 1386*sin(d*x
+ c) - 1260)/(a*d*sin(d*x + c)^11)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/13860*(2310*sin(d*x + c)^5 - 1980*sin(d*x + c)^4 - 3465*sin(d*x + c)^3 + 3080*sin(d*x + c)^2 + 1386*sin(d*x
+ c) - 1260)/(a*d*sin(d*x + c)^11)

Mupad [B] (verification not implemented)

Time = 10.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{6}-\frac {{\sin \left (c+d\,x\right )}^4}{7}-\frac {{\sin \left (c+d\,x\right )}^3}{4}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {\sin \left (c+d\,x\right )}{10}-\frac {1}{11}}{a\,d\,{\sin \left (c+d\,x\right )}^{11}} \]

[In]

int(cos(c + d*x)^7/(sin(c + d*x)^12*(a + a*sin(c + d*x))),x)

[Out]

(sin(c + d*x)/10 + (2*sin(c + d*x)^2)/9 - sin(c + d*x)^3/4 - sin(c + d*x)^4/7 + sin(c + d*x)^5/6 - 1/11)/(a*d*
sin(c + d*x)^11)