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

Optimal. Leaf size=129 \[ -\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d} \]

[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)

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Rubi [A]  time = 0.098505, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[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 2836

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*x)/b
)^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]

Rule 12

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

Rule 88

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]))

Rubi steps

\begin{align*} \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{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 \operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x^{15}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{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]  time = 0.233332, size = 86, normalized size = 0.67 \[ -\frac{a \csc ^{14}(c+d x) (9940 \sin (c+d x)+41860 \sin (3 (c+d x))+20020 \sin (5 (c+d x))+8580 \sin (7 (c+d x))+129129 \cos (2 (c+d x))+54054 \cos (4 (c+d x))+15015 \cos (6 (c+d x))+76362)}{3843840 d} \]

Antiderivative was successfully verified.

[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]^14*(76362 + 129129*Cos[2*(c + d*x)] + 54054*Cos[4*(c + d*x)] + 15015*Cos[6*(c + d*x)] + 9940*
Sin[c + d*x] + 41860*Sin[3*(c + d*x)] + 20020*Sin[5*(c + d*x)] + 8580*Sin[7*(c + d*x)]))/(3843840*d)

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Maple [B]  time = 0.063, size = 248, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{13\, \left ( \sin \left ( dx+c \right ) \right ) ^{13}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{143\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{429\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\,\sin \left ( dx+c \right ) }}+{\frac{5\,\sin \left ( dx+c \right ) }{3003} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{14\, \left ( \sin \left ( dx+c \right ) \right ) ^{14}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{28\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{70\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{280\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a*(-1/13/sin(d*x+c)^13*cos(d*x+c)^8-5/143/sin(d*x+c)^11*cos(d*x+c)^8-5/429/sin(d*x+c)^9*cos(d*x+c)^8-5/30
03/sin(d*x+c)^7*cos(d*x+c)^8+1/3003/sin(d*x+c)^5*cos(d*x+c)^8-1/3003/sin(d*x+c)^3*cos(d*x+c)^8+5/3003/sin(d*x+
c)*cos(d*x+c)^8+5/3003*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))+a*(-1/14/sin(d*x+c)^1
4*cos(d*x+c)^8-1/28/sin(d*x+c)^12*cos(d*x+c)^8-1/70/sin(d*x+c)^10*cos(d*x+c)^8-1/280/sin(d*x+c)^8*cos(d*x+c)^8
))

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Maxima [A]  time = 1.01359, size = 124, normalized size = 0.96 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.27976, size = 460, normalized size = 3.57 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.37425, size = 124, normalized size = 0.96 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)