3.222 \(\int \frac{1}{\csc (c+d x)-\sin (c+d x)} \, dx\)

Optimal. Leaf size=10 \[ \frac{\sec (c+d x)}{d} \]

[Out]

Sec[c + d*x]/d

________________________________________________________________________________________

Rubi [A]  time = 0.0250854, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4397, 2606, 8} \[ \frac{\sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x] - Sin[c + d*x])^(-1),x]

[Out]

Sec[c + d*x]/d

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

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

Mathematica [A]  time = 0.0091558, size = 10, normalized size = 1. \[ \frac{\sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]/d

________________________________________________________________________________________

Maple [A]  time = 0.076, size = 13, normalized size = 1.3 \begin{align*}{\frac{1}{d\cos \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.04778, size = 38, normalized size = 3.8 \begin{align*} -\frac{2}{d{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.449018, size = 27, normalized size = 2.7 \begin{align*} \frac{1}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(d*cos(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{- \sin{\left (c + d x \right )} + \csc{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(-sin(c + d*x) + csc(c + d*x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.10124, size = 38, normalized size = 3.8 \begin{align*} \frac{2}{d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))