\(\int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx\) [676]

Optimal result

Integrand size = 27, antiderivative size = 129 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^{10}(c+d x)}{10 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^{12}(c+d x)}{4 d}-\frac {a \csc ^{13}(c+d x)}{13 d}-\frac {a \csc ^{14}(c+d x)}{14 d} \]

[Out]

1/7*a*csc(d*x+c)^7/d+1/8*a*csc(d*x+c)^8/d-1/3*a*csc(d*x+c)^9/d-3/10*a*csc(d*x+c)^10/d+3/11*a*csc(d*x+c)^11/d+1
/4*a*csc(d*x+c)^12/d-1/13*a*csc(d*x+c)^13/d-1/14*a*csc(d*x+c)^14/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 129, 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 = {2915, 12, 90} \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^{14}(c+d x)}{14 d}-\frac {a \csc ^{13}(c+d x)}{13 d}+\frac {a \csc ^{12}(c+d x)}{4 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {3 a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {a \csc ^8(c+d x)}{8 d}+\frac {a \csc ^7(c+d x)}{7 d} \]

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^8*(a + a*Sin[c + d*x]),x]

[Out]

(a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(3*d) - (3*a*Csc[c + d*x]^10)/(10*d)
+ (3*a*Csc[c + d*x]^11)/(11*d) + (a*Csc[c + d*x]^12)/(4*d) - (a*Csc[c + d*x]^13)/(13*d) - (a*Csc[c + d*x]^14)/
(14*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^{15} (a-x)^3 (a+x)^4}{x^{15}} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {a^8 \text {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^{15}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \left (\frac {a^7}{x^{15}}+\frac {a^6}{x^{14}}-\frac {3 a^5}{x^{13}}-\frac {3 a^4}{x^{12}}+\frac {3 a^3}{x^{11}}+\frac {3 a^2}{x^{10}}-\frac {a}{x^9}-\frac {1}{x^8}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^{10}(c+d x)}{10 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^{12}(c+d x)}{4 d}-\frac {a \csc ^{13}(c+d x)}{13 d}-\frac {a \csc ^{14}(c+d x)}{14 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^{10}(c+d x)}{10 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^{12}(c+d x)}{4 d}-\frac {a \csc ^{13}(c+d x)}{13 d}-\frac {a \csc ^{14}(c+d x)}{14 d} \]

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^8*(a + a*Sin[c + d*x]),x]

[Out]

(a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(3*d) - (3*a*Csc[c + d*x]^10)/(10*d)
+ (3*a*Csc[c + d*x]^11)/(11*d) + (a*Csc[c + d*x]^12)/(4*d) - (a*Csc[c + d*x]^13)/(13*d) - (a*Csc[c + d*x]^14)/
(14*d)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{14}\left (d x +c \right )\right )}{14}+\frac {\left (\csc ^{13}\left (d x +c \right )\right )}{13}-\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{4}-\frac {3 \left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\left (\csc ^{14}\left (d x +c \right )\right )}{14}+\frac {\left (\csc ^{13}\left (d x +c \right )\right )}{13}-\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{4}-\frac {3 \left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}\right )}{d}\)	\(88\)
risch	\(-\frac {32 i a \left (15015 i {\mathrm e}^{20 i \left (d x +c \right )}+8580 \,{\mathrm e}^{21 i \left (d x +c \right )}+54054 i {\mathrm e}^{18 i \left (d x +c \right )}+20020 \,{\mathrm e}^{19 i \left (d x +c \right )}+129129 i {\mathrm e}^{16 i \left (d x +c \right )}+41860 \,{\mathrm e}^{17 i \left (d x +c \right )}+152724 i {\mathrm e}^{14 i \left (d x +c \right )}+9940 \,{\mathrm e}^{15 i \left (d x +c \right )}+129129 i {\mathrm e}^{12 i \left (d x +c \right )}-9940 \,{\mathrm e}^{13 i \left (d x +c \right )}+54054 i {\mathrm e}^{10 i \left (d x +c \right )}-41860 \,{\mathrm e}^{11 i \left (d x +c \right )}+15015 i {\mathrm e}^{8 i \left (d x +c \right )}-20020 \,{\mathrm e}^{9 i \left (d x +c \right )}-8580 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{15015 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{14}}\)	\(194\)
parallelrisch	\(-\frac {\left (\tan ^{28}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {28 \left (\tan ^{27}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{13}-\frac {28 \left (\tan ^{25}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {49 \left (\tan ^{24}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {56 \left (\tan ^{23}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+24 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 \left (\tan ^{20}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+84 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-245 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-560 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-560 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-245 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+84 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {56 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {49 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {28 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {28 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13}+1\right ) a}{229376 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}\)	\(289\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^15*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/14*csc(d*x+c)^14+1/13*csc(d*x+c)^13-1/4*csc(d*x+c)^12-3/11*csc(d*x+c)^11+3/10*csc(d*x+c)^10+1/3*csc(d*
x+c)^9-1/8*csc(d*x+c)^8-1/7*csc(d*x+c)^7)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\frac {15015 \, a \cos \left (d x + c\right )^{6} - 9009 \, a \cos \left (d x + c\right )^{4} + 3003 \, a \cos \left (d x + c\right )^{2} + 40 \, {\left (429 \, a \cos \left (d x + c\right )^{6} - 286 \, a \cos \left (d x + c\right )^{4} + 104 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 429 \, a}{120120 \, {\left (d \cos \left (d x + c\right )^{14} - 7 \, d \cos \left (d x + c\right )^{12} + 21 \, d \cos \left (d x + c\right )^{10} - 35 \, d \cos \left (d x + c\right )^{8} + 35 \, d \cos \left (d x + c\right )^{6} - 21 \, d \cos \left (d x + c\right )^{4} + 7 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^15*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/120120*(15015*a*cos(d*x + c)^6 - 9009*a*cos(d*x + c)^4 + 3003*a*cos(d*x + c)^2 + 40*(429*a*cos(d*x + c)^6 -
286*a*cos(d*x + c)^4 + 104*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 429*a)/(d*cos(d*x + c)^14 - 7*d*cos(d*x + c
)^12 + 21*d*cos(d*x + c)^10 - 35*d*cos(d*x + c)^8 + 35*d*cos(d*x + c)^6 - 21*d*cos(d*x + c)^4 + 7*d*cos(d*x +
c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**15*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\frac {17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^15*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/120120*(17160*a*sin(d*x + c)^7 + 15015*a*sin(d*x + c)^6 - 40040*a*sin(d*x + c)^5 - 36036*a*sin(d*x + c)^4 +
32760*a*sin(d*x + c)^3 + 30030*a*sin(d*x + c)^2 - 9240*a*sin(d*x + c) - 8580*a)/(d*sin(d*x + c)^14)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=\frac {17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^15*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/120120*(17160*a*sin(d*x + c)^7 + 15015*a*sin(d*x + c)^6 - 40040*a*sin(d*x + c)^5 - 36036*a*sin(d*x + c)^4 +
32760*a*sin(d*x + c)^3 + 30030*a*sin(d*x + c)^2 - 9240*a*sin(d*x + c) - 8580*a)/(d*sin(d*x + c)^14)

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{10}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{11}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{4}+\frac {a\,\sin \left (c+d\,x\right )}{13}+\frac {a}{14}}{d\,{\sin \left (c+d\,x\right )}^{14}} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^15,x)

[Out]

-(a/14 + (a*sin(c + d*x))/13 - (a*sin(c + d*x)^2)/4 - (3*a*sin(c + d*x)^3)/11 + (3*a*sin(c + d*x)^4)/10 + (a*s
in(c + d*x)^5)/3 - (a*sin(c + d*x)^6)/8 - (a*sin(c + d*x)^7)/7)/(d*sin(c + d*x)^14)