\(\int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx\) [1213]

Optimal result

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

[Out]

-1/5*b*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d+2/7*b*csc(d*x+c)^7/d+1/4*a*csc(d*x+c)^8/d-1/9*b*csc(d*x+c)^9/d-1/10
*a*csc(d*x+c)^10/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} \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \csc ^{10}(c+d x)}{10 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {b \csc ^9(c+d x)}{9 d}+\frac {2 b \csc ^7(c+d x)}{7 d}-\frac {b \csc ^5(c+d x)}{5 d} \]

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 b \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {b \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \]

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]

[Out]

-1/5*(b*Csc[c + d*x]^5)/d - (a*Csc[c + d*x]^6)/(6*d) + (2*b*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) -
(b*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{10}\left (d x +c \right )\right ) a}{10}+\frac {b \left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a}{4}-\frac {2 b \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{6}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(73\)
default	\(-\frac {\frac {\left (\csc ^{10}\left (d x +c \right )\right ) a}{10}+\frac {b \left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a}{4}-\frac {2 b \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{6}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(73\)
risch	\(-\frac {32 i \left (105 i a \,{\mathrm e}^{14 i \left (d x +c \right )}+63 b \,{\mathrm e}^{15 i \left (d x +c \right )}+210 i a \,{\mathrm e}^{12 i \left (d x +c \right )}+45 b \,{\mathrm e}^{13 i \left (d x +c \right )}+378 i a \,{\mathrm e}^{10 i \left (d x +c \right )}+110 b \,{\mathrm e}^{11 i \left (d x +c \right )}+210 i a \,{\mathrm e}^{8 i \left (d x +c \right )}-110 b \,{\mathrm e}^{9 i \left (d x +c \right )}+105 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-45 b \,{\mathrm e}^{7 i \left (d x +c \right )}-63 b \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\)	\(158\)
parallelrisch	\(\frac {-63 \left (\tan ^{20}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -140 b \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +525 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +1008 b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3150 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -7560 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -7560 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -3150 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1008 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+525 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +180 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-63 a}{645120 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}\)	\(229\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^11*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d*(1/10*csc(d*x+c)^10*a+1/9*b*csc(d*x+c)^9-1/4*csc(d*x+c)^8*a-2/7*b*csc(d*x+c)^7+1/6*csc(d*x+c)^6*a+1/5*b*c
sc(d*x+c)^5)

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=\frac {210 \, a \cos \left (d x + c\right )^{4} - 105 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, b \cos \left (d x + c\right )^{4} - 36 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 21 \, a}{1260 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/1260*(210*a*cos(d*x + c)^4 - 105*a*cos(d*x + c)^2 + 4*(63*b*cos(d*x + c)^4 - 36*b*cos(d*x + c)^2 + 8*b)*sin(
d*x + c) + 21*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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**11*(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 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, b \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, b \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1260*(252*b*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*b*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*b*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, b \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, b \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1260*(252*b*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*b*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*b*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

Mupad [B] (verification not implemented)

Time = 11.78 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252\,b\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-360\,b\,{\sin \left (c+d\,x\right )}^3-315\,a\,{\sin \left (c+d\,x\right )}^2+140\,b\,\sin \left (c+d\,x\right )+126\,a}{1260\,d\,{\sin \left (c+d\,x\right )}^{10}} \]

[In]

int((cos(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^11,x)

[Out]

-(126*a + 140*b*sin(c + d*x) - 315*a*sin(c + d*x)^2 + 210*a*sin(c + d*x)^4 - 360*b*sin(c + d*x)^3 + 252*b*sin(
c + d*x)^5)/(1260*d*sin(c + d*x)^10)