\(\int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx\) [1214]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {b \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {b \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2916, 12, 780} \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {b \csc ^{10}(c+d x)}{10 d}+\frac {b \csc ^8(c+d x)}{4 d}-\frac {b \csc ^6(c+d x)}{6 d} \]

[In]

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

[Out]

-1/6*(b*Csc[c + d*x]^6)/d - (a*Csc[c + d*x]^7)/(7*d) + (b*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) -
(b*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*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 780

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rule 2916

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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {b \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {b \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{11}\left (d x +c \right )\right ) a}{11}+\frac {b \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right ) a}{9}-\frac {b \left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right ) a}{7}+\frac {b \left (\csc ^{6}\left (d x +c \right )\right )}{6}}{d}\)	\(73\)
default	\(-\frac {\frac {\left (\csc ^{11}\left (d x +c \right )\right ) a}{11}+\frac {b \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right ) a}{9}-\frac {b \left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right ) a}{7}+\frac {b \left (\csc ^{6}\left (d x +c \right )\right )}{6}}{d}\)	\(73\)
parallelrisch	\(-\frac {5 \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )+\frac {9 \cos \left (4 d x +4 c \right )}{20}+\frac {37}{44}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {21 b \left (\cos \left (10 d x +10 c \right )+2770 \cos \left (2 d x +2 c \right )+1160 \cos \left (4 d x +4 c \right )+45 \cos \left (6 d x +6 c \right )-10 \cos \left (8 d x +8 c \right )+2178\right )}{25600}\right ) \left (\sec ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{258048 d}\)	\(129\)
risch	\(\frac {\frac {128 i a \,{\mathrm e}^{15 i \left (d x +c \right )}}{7}+\frac {32 b \,{\mathrm e}^{16 i \left (d x +c \right )}}{3}+\frac {2560 i a \,{\mathrm e}^{13 i \left (d x +c \right )}}{63}+\frac {32 b \,{\mathrm e}^{14 i \left (d x +c \right )}}{3}+\frac {47360 i a \,{\mathrm e}^{11 i \left (d x +c \right )}}{693}+\frac {256 b \,{\mathrm e}^{12 i \left (d x +c \right )}}{15}+\frac {2560 i a \,{\mathrm e}^{9 i \left (d x +c \right )}}{63}-\frac {256 b \,{\mathrm e}^{10 i \left (d x +c \right )}}{15}+\frac {128 i a \,{\mathrm e}^{7 i \left (d x +c \right )}}{7}-\frac {32 b \,{\mathrm e}^{8 i \left (d x +c \right )}}{3}-\frac {32 b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\)	\(157\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^12*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.32 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=\frac {1980 \, a \cos \left (d x + c\right )^{4} - 880 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, b \cos \left (d x + c\right )^{4} - 5 \, b \cos \left (d x + c\right )^{2} + b\right )} \sin \left (d x + c\right ) + 160 \, a}{13860 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**12*(a+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {2310 \, b \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, b \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, b \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {2310 \, b \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, b \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, b \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.62 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {\frac {b\,{\sin \left (c+d\,x\right )}^5}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {b\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {b\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]

[In]

int((cos(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^12,x)

[Out]

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