\(\int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx\) [222]

Optimal result

Integrand size = 17, antiderivative size = 10 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec (c+d x)}{d} \]

[Out]

sec(d*x+c)/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4482, 2686, 8} \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec (c+d x)}{d} \]

[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 2686

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 4482

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

Rubi steps \begin{align*} \text {integral}& = \int \sec (c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d} \\ & = \frac {\sec (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec (c+d x)}{d} \]

[In]

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

[Out]

Sec[c + d*x]/d

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {1}{d \cos \left (d x +c \right )}\)	\(13\)
default	\(\frac {1}{d \cos \left (d x +c \right )}\)	\(13\)
norman	\(-\frac {2}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(21\)
parallelrisch	\(-\frac {2}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(21\)
risch	\(\frac {2 \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {1}{d \cos \left (d x + c\right )} \]

[In]

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

[Out]

1/(d*cos(d*x + c))

Sympy [F]

\[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\int \frac {1}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (10) = 20\).

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}} \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {2}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 24.59 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

-2/(d*(tan(c/2 + (d*x)/2)^2 - 1))