3.252 \(\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=30 \[ \frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]

[Out]

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

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Rubi [A]  time = 0.065052, antiderivative size = 30, normalized size of antiderivative = 1., 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 = {2833, 12, 37} \[ \frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

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]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\sin ^3(c+d x)}{3 a d (a+a \sin (c+d x))^3}\\ \end{align*}

Mathematica [A]  time = 0.183911, size = 53, normalized size = 1.77 \[ \frac{-6 \sin (c+d x)+3 \cos (2 (c+d x))-5}{6 a^4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]

[Out]

(-5 + 3*Cos[2*(c + d*x)] - 6*Sin[c + d*x])/(6*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.034, size = 43, normalized size = 1.4 \begin{align*}{\frac{1}{d{a}^{4}} \left ( \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-2}-{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}- \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a^4*(1/(1+sin(d*x+c))^2-1/3/(1+sin(d*x+c))^3-1/(1+sin(d*x+c)))

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Maxima [B]  time = 1.08461, size = 90, normalized size = 3. \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \,{\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/3*(3*sin(d*x + c)^2 + 3*sin(d*x + c) + 1)/((a^4*sin(d*x + c)^3 + 3*a^4*sin(d*x + c)^2 + 3*a^4*sin(d*x + c)
+ a^4)*d)

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Fricas [B]  time = 1.36662, size = 174, normalized size = 5.8 \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) - 4}{3 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*(3*cos(d*x + c)^2 - 3*sin(d*x + c) - 4)/(3*a^4*d*cos(d*x + c)^2 - 4*a^4*d + (a^4*d*cos(d*x + c)^2 - 4*a^4
*d)*sin(d*x + c))

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Sympy [A]  time = 3.98311, size = 76, normalized size = 2.53 \begin{align*} \begin{cases} \frac{\sin ^{3}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin{\left (c + d x \right )} + 3 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{2}{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)**2/(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((sin(c + d*x)**3/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a*
*4*d), Ne(d, 0)), (x*sin(c)**2*cos(c)/(a*sin(c) + a)**4, True))

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Giac [A]  time = 1.18622, size = 51, normalized size = 1.7 \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{4} d{\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*sin(d*x + c)^2 + 3*sin(d*x + c) + 1)/(a^4*d*(sin(d*x + c) + 1)^3)