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

Optimal. Leaf size=73 \[ -\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d} \]

[Out]

Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + (3*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(6*a^3
*d)

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Rubi [A]  time = 0.0574368, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 43} \[ -\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + (3*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(6*a^3
*d)

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

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

Mathematica [A]  time = 0.098613, size = 48, normalized size = 0.66 \[ \frac{\csc ^3(c+d x) \left (-10 \csc ^3(c+d x)+36 \csc ^2(c+d x)-45 \csc (c+d x)+20\right )}{60 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^3*(20 - 45*Csc[c + d*x] + 36*Csc[c + d*x]^2 - 10*Csc[c + d*x]^3))/(60*a^3*d)

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Maple [A]  time = 0.119, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{3}} \left ({\frac{3}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.53887, size = 62, normalized size = 0.85 \begin{align*} \frac{20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(20*sin(d*x + c)^3 - 45*sin(d*x + c)^2 + 36*sin(d*x + c) - 10)/(a^3*d*sin(d*x + c)^6)

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Fricas [A]  time = 1.46143, size = 208, normalized size = 2.85 \begin{align*} -\frac{45 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (5 \, \cos \left (d x + c\right )^{2} - 14\right )} \sin \left (d x + c\right ) - 55}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(45*cos(d*x + c)^2 - 4*(5*cos(d*x + c)^2 - 14)*sin(d*x + c) - 55)/(a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*
x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*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(cot(d*x+c)**7/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.64026, size = 62, normalized size = 0.85 \begin{align*} \frac{20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(20*sin(d*x + c)^3 - 45*sin(d*x + c)^2 + 36*sin(d*x + c) - 10)/(a^3*d*sin(d*x + c)^6)