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

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

[Out]

Sec[c + d*x]/(a*d)

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Rubi [A]  time = 0.0338714, antiderivative size = 13, 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, 8} \[ \frac{\sec (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[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 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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0127608, size = 13, normalized size = 1. \[ \frac{\sec (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]/(a*d)

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Maple [A]  time = 0.03, size = 16, normalized size = 1.2 \begin{align*}{\frac{1}{da\cos \left ( dx+c \right ) }} \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),x)

[Out]

1/d/a/cos(d*x+c)

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Maxima [A]  time = 0.949293, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \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),x, algorithm="maxima")

[Out]

1/(a*d*cos(d*x + c))

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Fricas [A]  time = 1.65355, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \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),x, algorithm="fricas")

[Out]

1/(a*d*cos(d*x + c))

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Sympy [A]  time = 3.1869, size = 34, normalized size = 2.62 \begin{align*} \begin{cases} - \frac{2}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )}}{- 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(sin(d*x+c)/(a-a*sin(d*x+c)**2),x)

[Out]

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

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Giac [A]  time = 1.11697, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \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),x, algorithm="giac")

[Out]

1/(a*d*cos(d*x + c))