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

Optimal. Leaf size=32 \[ \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2706, 2606, 30, 8} \[ \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 24, normalized size = 0.75 \[ -\frac {(\csc (c+d x)-2) \csc (c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-2 + Csc[c + d*x])*Csc[c + d*x])/(a*d)

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fricas [A]  time = 0.47, size = 30, normalized size = 0.94 \[ -\frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 26, normalized size = 0.81 \[ \frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.23, size = 25, normalized size = 0.78 \[ \frac {-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 26, normalized size = 0.81 \[ \frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 8.63, size = 23, normalized size = 0.72 \[ \frac {\sin \left (c+d\,x\right )-\frac {1}{2}}{a\,d\,{\sin \left (c+d\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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