3.52 \(\int \frac{\sin (c+d x)}{(a-a \sin ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]

[Out]

Sec[c + d*x]^3/(3*a^2*d)

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Rubi [A]  time = 0.0421728, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3175, 2606, 30} \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]^3/(3*a^2*d)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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 30

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

Rubi steps

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

Mathematica [A]  time = 0.0111101, size = 18, normalized size = 1. \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]^3/(3*a^2*d)

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Maple [A]  time = 0.032, size = 17, normalized size = 0.9 \begin{align*}{\frac{1}{3\,{a}^{2}d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.952217, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(a^2*d*cos(d*x + c)^3)

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Fricas [A]  time = 1.59587, size = 38, normalized size = 2.11 \begin{align*} \frac{1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(a^2*d*cos(d*x + c)^3)

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Sympy [A]  time = 19.495, size = 156, normalized size = 8.67 \begin{align*} \begin{cases} - \frac{6 \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 3 a^{2} d} - \frac{2}{3 a^{2} d \tan ^{6}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 3 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*tan(c/2 + d*x/2)**4/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(
c/2 + d*x/2)**2 - 3*a**2*d) - 2/(3*a**2*d*tan(c/2 + d*x/2)**6 - 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/
2 + d*x/2)**2 - 3*a**2*d), Ne(d, 0)), (x*sin(c)/(-a*sin(c)**2 + a)**2, True))

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Giac [A]  time = 1.15784, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(a^2*d*cos(d*x + c)^3)