3.45 \(\int \frac{1}{a-a \sin ^2(c+d x)} \, dx\)

Optimal. Leaf size=13 \[ \frac{\tan (c+d x)}{a d} \]

[Out]

Tan[c + d*x]/(a*d)

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Rubi [A]  time = 0.0224128, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3175, 3767, 8} \[ \frac{\tan (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[c + d*x]^2)^(-1),x]

[Out]

Tan[c + d*x]/(a*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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.00612, size = 13, normalized size = 1. \[ \frac{\tan (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[c + d*x]^2)^(-1),x]

[Out]

Tan[c + d*x]/(a*d)

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Maple [A]  time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(d*x+c)/d/a

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Maxima [A]  time = 0.947621, size = 18, normalized size = 1.38 \begin{align*} \frac{\tan \left (d x + c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

tan(d*x + c)/(a*d)

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Fricas [A]  time = 1.59973, size = 45, normalized size = 3.46 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

sin(d*x + c)/(a*d*cos(d*x + c))

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Sympy [A]  time = 2.67555, size = 41, normalized size = 3.15 \begin{align*} \begin{cases} - \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} & \text{for}\: d \neq 0 \\\frac{x}{- a \sin ^{2}{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12077, size = 18, normalized size = 1.38 \begin{align*} \frac{\tan \left (d x + c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(d*x+c)^2),x, algorithm="giac")

[Out]

tan(d*x + c)/(a*d)