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]

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

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Rubi [A]  time = 0.08, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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])

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 {\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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.48, size = 38, normalized size = 1.03 \[ -\frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.17, size = 26, normalized size = 0.70 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, a d \sin \left (d x + c\right )^{3}} \]

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)

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maple [A]  time = 0.20, size = 29, normalized size = 0.78 \[ \frac {-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{a d} \]

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

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maxima [A]  time = 0.31, size = 26, normalized size = 0.70 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{6 \, a d \sin \left (d x + c\right )^{3}} \]

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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mupad [B]  time = 8.76, size = 36, normalized size = 0.97 \[ \frac {\frac {5\,\sin \left (c+d\,x\right )}{16}+\frac {\sin \left (3\,c+3\,d\,x\right )}{16}-\frac {1}{3}}{a\,d\,{\sin \left (c+d\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*sin(c + d*x))/16 + sin(3*c + 3*d*x)/16 - 1/3)/(a*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)**3*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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