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

Optimal. Leaf size=37 \[ \frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]

[Out]

Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d)

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Rubi [A]  time = 0.0809194, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 43} \[ \frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0529001, size = 28, normalized size = 0.76 \[ \frac{(3 \sin (c+d x)-2) \csc ^3(c+d x)}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^3*(-2 + 3*Sin[c + d*x]))/(6*a*d)

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Maple [A]  time = 0.038, size = 29, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*csc(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.12923, size = 35, normalized size = 0.95 \begin{align*} \frac{3 \, \sin \left (d x + c\right ) - 2}{6 \, a 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)^3*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/6*(3*sin(d*x + c) - 2)/(a*d*sin(d*x + c)^3)

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Fricas [A]  time = 1.28677, size = 93, normalized size = 2.51 \begin{align*} -\frac{3 \, \sin \left (d x + c\right ) - 2}{6 \,{\left (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)^3*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(3*sin(d*x + c) - 2)/((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)**3*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.39771, size = 35, normalized size = 0.95 \begin{align*} \frac{3 \, \sin \left (d x + c\right ) - 2}{6 \, a 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)^3*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/6*(3*sin(d*x + c) - 2)/(a*d*sin(d*x + c)^3)