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(d*x+c)/d

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 8

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

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 4397

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

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*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {\sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]/d

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fricas [A]  time = 0.46, size = 12, normalized size = 1.20 \[ \frac {1}{d \cos \left (d x + c\right )} \]

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))

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giac [B]  time = 0.19, size = 28, normalized size = 2.80 \[ \frac {2}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} \]

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))

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maple [A]  time = 0.12, size = 13, normalized size = 1.30 \[ \frac {1}{d \cos \left (d x +c \right )} \]

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)

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maxima [B]  time = 0.42, size = 28, normalized size = 2.80 \[ -\frac {2}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}} \]

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))

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mupad [B]  time = 0.66, size = 20, normalized size = 2.00 \[ -\frac {2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(d*(tan(c/2 + (d*x)/2)^2 - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

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)

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