3.694 \(\int \frac{\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d} \]

[Out]

Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*d) - (2*Csc[c + d*x]^7)/(7*a*d) + Csc[c + d*x]^8/(4*a*d) + Csc[c
+ d*x]^9/(9*a*d) - Csc[c + d*x]^10/(10*a*d)

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Rubi [A]  time = 0.121184, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.106386, size = 68, normalized size = 0.62 \[ \frac{\csc ^5(c+d x) \left (-126 \csc ^5(c+d x)+140 \csc ^4(c+d x)+315 \csc ^3(c+d x)-360 \csc ^2(c+d x)-210 \csc (c+d x)+252\right )}{1260 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^5*(252 - 210*Csc[c + d*x] - 360*Csc[c + d*x]^2 + 315*Csc[c + d*x]^3 + 140*Csc[c + d*x]^4 - 126*C
sc[c + d*x]^5))/(1260*a*d)

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Maple [A]  time = 0.19, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ( -{\frac{1}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{2}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{6\, \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)^11/(a+a*sin(d*x+c)),x)

[Out]

1/d/a*(-1/10/sin(d*x+c)^10-2/7/sin(d*x+c)^7+1/4/sin(d*x+c)^8+1/5/sin(d*x+c)^5+1/9/sin(d*x+c)^9-1/6/sin(d*x+c)^
6)

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Maxima [A]  time = 0.995203, size = 89, normalized size = 0.82 \begin{align*} \frac{252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a 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]

1/1260*(252*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 360*sin(d*x + c)^3 + 315*sin(d*x + c)^2 + 140*sin(d*x + c) -
126)/(a*d*sin(d*x + c)^10)

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Fricas [A]  time = 1.13609, size = 321, normalized size = 2.94 \begin{align*} \frac{210 \, \cos \left (d x + c\right )^{4} - 105 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (63 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 21}{1260 \,{\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a 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]

1/1260*(210*cos(d*x + c)^4 - 105*cos(d*x + c)^2 - 4*(63*cos(d*x + c)^4 - 36*cos(d*x + c)^2 + 8)*sin(d*x + c) +
21)/(a*d*cos(d*x + c)^10 - 5*a*d*cos(d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*cos(d
*x + c)^2 - a*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 = 1.41387, size = 89, normalized size = 0.82 \begin{align*} \frac{252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a 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]

1/1260*(252*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 360*sin(d*x + c)^3 + 315*sin(d*x + c)^2 + 140*sin(d*x + c) -
126)/(a*d*sin(d*x + c)^10)