3.550 \(\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=31 \[ -\frac{\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d} \]

[Out]

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

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Rubi [A]  time = 0.079296, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 37} \[ -\frac{\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

-(Csc[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(3*a^5*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0440705, size = 20, normalized size = 0.65 \[ -\frac{(\csc (c+d x)-1)^3}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

-(-1 + Csc[c + d*x])^3/(3*a^2*d)

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Maple [A]  time = 0.132, size = 37, normalized size = 1.2 \begin{align*}{\frac{1}{d{a}^{2}} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+ \left ( \sin \left ( dx+c \right ) \right ) ^{-2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x)

[Out]

1/d/a^2*(-1/sin(d*x+c)-1/3/sin(d*x+c)^3+1/sin(d*x+c)^2)

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Maxima [A]  time = 1.05884, size = 49, normalized size = 1.58 \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(3*sin(d*x + c)^2 - 3*sin(d*x + c) + 1)/(a^2*d*sin(d*x + c)^3)

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Fricas [A]  time = 1.00157, size = 124, normalized size = 4. \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 4}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} 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)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3*(3*cos(d*x + c)^2 + 3*sin(d*x + c) - 4)/((a^2*d*cos(d*x + c)^2 - a^2*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)**5*csc(d*x+c)**4/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.22028, size = 49, normalized size = 1.58 \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*(3*sin(d*x + c)^2 - 3*sin(d*x + c) + 1)/(a^2*d*sin(d*x + c)^3)