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]

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

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Rubi [A]  time = 0.08, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.04, 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]

-1/3*(-1 + Csc[c + d*x])^3/(a^2*d)

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fricas [A]  time = 0.63, size = 52, normalized size = 1.68 \[ -\frac {3 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 4}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.23, size = 36, normalized size = 1.16 \[ -\frac {3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \]

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)

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maple [A]  time = 0.54, size = 37, normalized size = 1.19 \[ \frac {-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{\sin \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}} \]

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

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maxima [A]  time = 0.31, size = 36, normalized size = 1.16 \[ -\frac {3 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d \sin \left (d x + c\right )^{3}} \]

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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mupad [B]  time = 8.93, size = 34, normalized size = 1.10 \[ -\frac {{\sin \left (c+d\,x\right )}^2-\sin \left (c+d\,x\right )+\frac {1}{3}}{a^2\,d\,{\sin \left (c+d\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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