3.693 \(\int \frac{\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=91 \[ \frac{\cot ^8(c+d x)}{8 a d}+\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^9(c+d x)}{9 a d}+\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^5(c+d x)}{5 a d} \]

[Out]

Cot[c + d*x]^6/(6*a*d) + Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]^5/(5*a*d) + (2*Csc[c + d*x]^7)/(7*a*d) - Csc[c
+ d*x]^9/(9*a*d)

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Rubi [A]  time = 0.160435, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2835, 2606, 270, 2607, 14} \[ \frac{\cot ^8(c+d x)}{8 a d}+\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^9(c+d x)}{9 a d}+\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^7*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

Cot[c + d*x]^6/(6*a*d) + Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]^5/(5*a*d) + (2*Csc[c + d*x]^7)/(7*a*d) - Csc[c
+ d*x]^9/(9*a*d)

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

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 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]]

Rubi steps

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

Mathematica [A]  time = 0.155548, size = 68, normalized size = 0.75 \[ \frac{\csc ^4(c+d x) \left (-280 \csc ^5(c+d x)+315 \csc ^4(c+d x)+720 \csc ^3(c+d x)-840 \csc ^2(c+d x)-504 \csc (c+d x)+630\right )}{2520 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(Csc[c + d*x]^4*(630 - 504*Csc[c + d*x] - 840*Csc[c + d*x]^2 + 720*Csc[c + d*x]^3 + 315*Csc[c + d*x]^4 - 280*C
sc[c + d*x]^5))/(2520*a*d)

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Maple [A]  time = 0.177, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ({\frac{2}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x)

[Out]

1/d/a*(2/7/sin(d*x+c)^7+1/8/sin(d*x+c)^8-1/5/sin(d*x+c)^5+1/4/sin(d*x+c)^4-1/9/sin(d*x+c)^9-1/3/sin(d*x+c)^6)

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Maxima [A]  time = 1.01303, size = 89, normalized size = 0.98 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720*sin(d*x + c)^2 + 315*sin(d*x + c) -
280)/(a*d*sin(d*x + c)^9)

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Fricas [A]  time = 1.13092, size = 308, normalized size = 3.38 \begin{align*} -\frac{504 \, \cos \left (d x + c\right )^{4} - 288 \, \cos \left (d x + c\right )^{2} - 105 \,{\left (6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 64}{2520 \,{\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a 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)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(504*cos(d*x + c)^4 - 288*cos(d*x + c)^2 - 105*(6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*sin(d*x + c)
+ 64)/((a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*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)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.20144, size = 89, normalized size = 0.98 \begin{align*} \frac{630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2520*(630*sin(d*x + c)^5 - 504*sin(d*x + c)^4 - 840*sin(d*x + c)^3 + 720*sin(d*x + c)^2 + 315*sin(d*x + c) -
280)/(a*d*sin(d*x + c)^9)