3.670 \(\int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=74 \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.107246, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2607, 30, 2606, 194} \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) + (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c
+ d*x]^7)/(7*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 194

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

Rubi steps

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

Mathematica [A]  time = 0.0328331, size = 74, normalized size = 1. \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

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

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Maple [B]  time = 0.068, size = 138, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\,\sin \left ( dx+c \right ) }}+{\frac{\sin \left ( dx+c \right ) }{7} \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 ) -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/7/sin(d*x+c)^7*cos(d*x+c)^8+1/35/sin(d*x+c)^5*cos(d*x+c)^8-1/35/sin(d*x+c)^3*cos(d*x+c)^8+1/7/sin(d
*x+c)*cos(d*x+c)^8+1/7*(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))-1/8*a/sin(d*x+c)^8*co
s(d*x+c)^8)

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Maxima [A]  time = 1.04174, size = 124, normalized size = 1.68 \begin{align*} \frac{280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d
*x + c)^3 + 140*a*sin(d*x + c)^2 - 40*a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)

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Fricas [A]  time = 1.13352, size = 351, normalized size = 4.74 \begin{align*} -\frac{140 \, a \cos \left (d x + c\right )^{6} - 210 \, a \cos \left (d x + c\right )^{4} + 140 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (35 \, a \cos \left (d x + c\right )^{6} - 70 \, a \cos \left (d x + c\right )^{4} + 56 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 35 \, a}{280 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, 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)^9*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/280*(140*a*cos(d*x + c)^6 - 210*a*cos(d*x + c)^4 + 140*a*cos(d*x + c)^2 + 8*(35*a*cos(d*x + c)^6 - 70*a*cos
(d*x + c)^4 + 56*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 35*a)/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*co
s(d*x + c)^4 - 4*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)**9*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.35942, size = 124, normalized size = 1.68 \begin{align*} \frac{280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d
*x + c)^3 + 140*a*sin(d*x + c)^2 - 40*a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)