3.2 \(\int \frac {\cos ^3(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {\sin ^2(x)}{2 a}-\frac {\sin ^3(x)}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3872, 2835, 2564, 30} \[ \frac {\sin ^2(x)}{2 a}-\frac {\sin ^3(x)}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3/(a + a*Csc[x]),x]

[Out]

Sin[x]^2/(2*a) - Sin[x]^3/(3*a)

Rule 30

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

Rule 2564

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

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.74 \[ \frac {(3-2 \sin (x)) \sin ^2(x)}{6 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3/(a + a*Csc[x]),x]

[Out]

((3 - 2*Sin[x])*Sin[x]^2)/(6*a)

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fricas [A]  time = 0.64, size = 22, normalized size = 0.96 \[ -\frac {3 \, \cos \relax (x)^{2} - 2 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*cos(x)^2 - 2*(cos(x)^2 - 1)*sin(x))/a

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giac [A]  time = 0.43, size = 18, normalized size = 0.78 \[ -\frac {2 \, \sin \relax (x)^{3} - 3 \, \sin \relax (x)^{2}}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*sin(x)^3 - 3*sin(x)^2)/a

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maple [A]  time = 0.23, size = 18, normalized size = 0.78 \[ \frac {-\frac {\left (\sin ^{3}\relax (x )\right )}{3}+\frac {\left (\sin ^{2}\relax (x )\right )}{2}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 18, normalized size = 0.78 \[ -\frac {2 \, \sin \relax (x)^{3} - 3 \, \sin \relax (x)^{2}}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*sin(x)^3 - 3*sin(x)^2)/a

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mupad [B]  time = 0.04, size = 15, normalized size = 0.65 \[ -\frac {{\sin \relax (x)}^2\,\left (2\,\sin \relax (x)-3\right )}{6\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3/(a + a/sin(x)),x)

[Out]

-(sin(x)^2*(2*sin(x) - 3))/(6*a)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos ^{3}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3/(a+a*csc(x)),x)

[Out]

Integral(cos(x)**3/(csc(x) + 1), x)/a

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