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

Optimal. Leaf size=113 \[ -\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^7(c+d x)}{7 d} \]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^7)/(7*d)
- (a*Csc[c + d*x]^9)/(3*d) + (3*a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^13)/(13*d)

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Rubi [A]  time = 0.134878, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2834, 2606, 270, 2607, 266, 43} \[ -\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^7)/(7*d)
- (a*Csc[c + d*x]^9)/(3*d) + (3*a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^13)/(13*d)

Rule 2834

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^6(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^7(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int x^7 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int x^3 (1+x)^2 \, dx,x,\cot ^2(c+d x)\right )}{2 d}-\frac{a \operatorname{Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{13}(c+d x)}{13 d}-\frac{a \operatorname{Subst}\left (\int \left (x^3+2 x^4+x^5\right ) \, dx,x,\cot ^2(c+d x)\right )}{2 d}\\ &=-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^{12}(c+d x)}{12 d}+\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{13}(c+d x)}{13 d}\\ \end{align*}

Mathematica [A]  time = 0.215459, size = 86, normalized size = 0.76 \[ -\frac{a \csc ^{13}(c+d x) (3003 \sin (c+d x)+24024 \sin (3 (c+d x))+10010 \sin (5 (c+d x))+5005 \sin (7 (c+d x))+70460 \cos (2 (c+d x))+28600 \cos (4 (c+d x))+8580 \cos (6 (c+d x))+40200)}{1921920 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Csc[c + d*x]^13*(40200 + 70460*Cos[2*(c + d*x)] + 28600*Cos[4*(c + d*x)] + 8580*Cos[6*(c + d*x)] + 3003*Si
n[c + d*x] + 24024*Sin[3*(c + d*x)] + 10010*Sin[5*(c + d*x)] + 5005*Sin[7*(c + d*x)]))/(1921920*d)

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Maple [B]  time = 0.066, size = 230, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{12\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{30\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{120\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) +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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^14*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/12/sin(d*x+c)^12*cos(d*x+c)^8-1/30/sin(d*x+c)^10*cos(d*x+c)^8-1/120/sin(d*x+c)^8*cos(d*x+c)^8)+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)^8-5/3003/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)))

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Maxima [A]  time = 1.05411, size = 124, normalized size = 1.1 \begin{align*} \frac{20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^14*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/120120*(20020*a*sin(d*x + c)^7 + 17160*a*sin(d*x + c)^6 - 45045*a*sin(d*x + c)^5 - 40040*a*sin(d*x + c)^4 +
36036*a*sin(d*x + c)^3 + 32760*a*sin(d*x + c)^2 - 10010*a*sin(d*x + c) - 9240*a)/(d*sin(d*x + c)^13)

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Fricas [A]  time = 1.25785, size = 446, normalized size = 3.95 \begin{align*} -\frac{17160 \, a \cos \left (d x + c\right )^{6} - 11440 \, a \cos \left (d x + c\right )^{4} + 4160 \, a \cos \left (d x + c\right )^{2} + 1001 \,{\left (20 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 640 \, a}{120120 \,{\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^14*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/120120*(17160*a*cos(d*x + c)^6 - 11440*a*cos(d*x + c)^4 + 4160*a*cos(d*x + c)^2 + 1001*(20*a*cos(d*x + c)^6
- 15*a*cos(d*x + c)^4 + 6*a*cos(d*x + c)^2 - a)*sin(d*x + c) - 640*a)/((d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^
10 + 15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d*x + c)^2 + d)*sin(d*x + c))

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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)**14*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33499, size = 124, normalized size = 1.1 \begin{align*} \frac{20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^14*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/120120*(20020*a*sin(d*x + c)^7 + 17160*a*sin(d*x + c)^6 - 45045*a*sin(d*x + c)^5 - 40040*a*sin(d*x + c)^4 +
36036*a*sin(d*x + c)^3 + 32760*a*sin(d*x + c)^2 - 10010*a*sin(d*x + c) - 9240*a)/(d*sin(d*x + c)^13)