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(d*x+c)/d

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Rubi [A]  time = 0.09, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\tan (c+d x)}{d} \]

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

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fricas [A]  time = 0.42, size = 18, normalized size = 1.80 \[ \frac {\sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]

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))

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giac [A]  time = 0.20, size = 10, normalized size = 1.00 \[ \frac {\tan \left (d x + c\right )}{d} \]

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

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maple [A]  time = 0.11, size = 11, normalized size = 1.10 \[ \frac {\tan \left (d x +c \right )}{d} \]

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

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maxima [B]  time = 0.34, size = 44, normalized size = 4.40 \[ -\frac {2 \, \sin \left (d x + c\right )}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} \]

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))

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mupad [B]  time = 0.58, size = 29, normalized size = 2.90 \[ -\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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