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

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

[Out]

Tan[c + d*x]/d

________________________________________________________________________________________

Rubi [A]  time = 0.0862574, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {8} \[ \frac{\tan (c+d x)}{d} \]

Antiderivative was successfully verified.

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

Mathematica [A]  time = 0.0039757, size = 10, normalized size = 1. \[ \frac{\tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[c + d*x]/d

________________________________________________________________________________________

Maple [A]  time = 0.067, size = 11, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(d*x+c)/d

________________________________________________________________________________________

Maxima [B]  time = 1.05784, size = 59, normalized size = 5.9 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [A]  time = 0.447982, size = 42, normalized size = 4.2 \begin{align*} \frac{\sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Giac [A]  time = 1.14395, size = 14, normalized size = 1.4 \begin{align*} \frac{\tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(d*x + c)/d