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

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

[Out]

Tan[c + d*x]^3/(3*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0674126, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2607, 30} \[ \frac{\tan ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[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 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 ^2(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0176636, size = 18, normalized size = 1. \[ \frac{\tan ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[c + d*x]^3/(3*a^2*d)

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 17, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.13808, size = 22, normalized size = 1.22 \begin{align*} \frac{\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(d*x + c)^3/(a^2*d)

________________________________________________________________________________________

Fricas [A]  time = 1.53549, size = 85, normalized size = 4.72 \begin{align*} -\frac{{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{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)^2/(a-a*sin(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 31.7744, size = 94, normalized size = 5.22 \begin{align*} \begin{cases} - \frac{8 \tan ^{3}{\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} & \text{for}\: d \neq 0 \\\frac{x \sin ^{2}{\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)**2/(a-a*sin(d*x+c)**2)**2,x)

[Out]

Piecewise((-8*tan(c/2 + d*x/2)**3/(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)**2/(-a*sin(c)**2 + a)**2, True))

________________________________________________________________________________________

Giac [A]  time = 1.1213, size = 22, normalized size = 1.22 \begin{align*} \frac{\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(d*x + c)^3/(a^2*d)