3.672 \(\int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=97 \[ -\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.125265, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2607, 14, 2606, 270} \[ -\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d)
+ (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*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 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

Rubi steps

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

Mathematica [A]  time = 0.184479, size = 86, normalized size = 0.89 \[ -\frac{a \csc ^3(c+d x) \left (252 \csc ^7(c+d x)+280 \csc ^6(c+d x)-945 \csc ^5(c+d x)-1080 \csc ^4(c+d x)+1260 \csc ^3(c+d x)+1512 \csc ^2(c+d x)-630 \csc (c+d x)-840\right )}{2520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^3*(-840 - 630*Csc[c + d*x] + 1512*Csc[c + d*x]^2 + 1260*Csc[c + d*x]^3 - 1080*Csc[c + d*x]^4
- 945*Csc[c + d*x]^5 + 280*Csc[c + d*x]^6 + 252*Csc[c + d*x]^7))/(2520*d)

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Maple [B]  time = 0.062, size = 176, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{315\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{315\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{63\,\sin \left ( dx+c \right ) }}+{\frac{\sin \left ( dx+c \right ) }{63} \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}}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{40\, \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)^11*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/9/sin(d*x+c)^9*cos(d*x+c)^8-1/63/sin(d*x+c)^7*cos(d*x+c)^8+1/315/sin(d*x+c)^5*cos(d*x+c)^8-1/315/si
n(d*x+c)^3*cos(d*x+c)^8+1/63/sin(d*x+c)*cos(d*x+c)^8+1/63*(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*(-1/10/sin(d*x+c)^10*cos(d*x+c)^8-1/40/sin(d*x+c)^8*cos(d*x+c)^8))

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Maxima [A]  time = 1.03559, size = 124, normalized size = 1.28 \begin{align*} \frac{840 \, a \sin \left (d x + c\right )^{7} + 630 \, a \sin \left (d x + c\right )^{6} - 1512 \, a \sin \left (d x + c\right )^{5} - 1260 \, a \sin \left (d x + c\right )^{4} + 1080 \, a \sin \left (d x + c\right )^{3} + 945 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right ) - 252 \, a}{2520 \, d \sin \left (d x + c\right )^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2520*(840*a*sin(d*x + c)^7 + 630*a*sin(d*x + c)^6 - 1512*a*sin(d*x + c)^5 - 1260*a*sin(d*x + c)^4 + 1080*a*s
in(d*x + c)^3 + 945*a*sin(d*x + c)^2 - 280*a*sin(d*x + c) - 252*a)/(d*sin(d*x + c)^10)

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Fricas [A]  time = 1.16464, size = 386, normalized size = 3.98 \begin{align*} \frac{630 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (105 \, a \cos \left (d x + c\right )^{6} - 126 \, a \cos \left (d x + c\right )^{4} + 72 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 63 \, a}{2520 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2520*(630*a*cos(d*x + c)^6 - 630*a*cos(d*x + c)^4 + 315*a*cos(d*x + c)^2 + 8*(105*a*cos(d*x + c)^6 - 126*a*c
os(d*x + c)^4 + 72*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 63*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)

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

[Out]

Timed out

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Giac [A]  time = 1.29762, size = 124, normalized size = 1.28 \begin{align*} \frac{840 \, a \sin \left (d x + c\right )^{7} + 630 \, a \sin \left (d x + c\right )^{6} - 1512 \, a \sin \left (d x + c\right )^{5} - 1260 \, a \sin \left (d x + c\right )^{4} + 1080 \, a \sin \left (d x + c\right )^{3} + 945 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right ) - 252 \, a}{2520 \, d \sin \left (d x + c\right )^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2520*(840*a*sin(d*x + c)^7 + 630*a*sin(d*x + c)^6 - 1512*a*sin(d*x + c)^5 - 1260*a*sin(d*x + c)^4 + 1080*a*s
in(d*x + c)^3 + 945*a*sin(d*x + c)^2 - 280*a*sin(d*x + c) - 252*a)/(d*sin(d*x + c)^10)