\(\int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\) [228]

Optimal result

Integrand size = 24, antiderivative size = 10 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\tan (c+d x)}{d} \]

[Out]

tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {8} \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\tan (c+d x)}{d} \]

[In]

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

[Out]

Tan[c + d*x]/d

Rule 8

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\tan (c+d x)}{d} \]

[In]

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

[Out]

Tan[c + d*x]/d

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {\tan \left (d x +c \right )}{d}\)	\(11\)
default	\(\frac {\tan \left (d x +c \right )}{d}\)	\(11\)
risch	\(\frac {2 i}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\)	\(20\)
norman	\(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(30\)
parallelrisch	\(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(30\)

[In]

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

[Out]

tan(d*x+c)/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]

[In]

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

[Out]

sin(d*x + c)/(d*cos(d*x + c))

Sympy [F]

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

[In]

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

[Out]

Integral(csc(c + d*x)/(-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. 44 vs. \(2 (10) = 20\).

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 4.40 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2 \, \sin \left (d x + c\right )}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\tan \left (d x + c\right )}{d} \]

[In]

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

[Out]

tan(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 22.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.90 \[ \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

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