\(\int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\) [223]

Optimal result

Integrand size = 24, antiderivative size = 14 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-x+\frac {\tan (c+d x)}{d} \]

[Out]

-x+tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {327, 209} \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\tan (c+d x)}{d}-x \]

[In]

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

[Out]

-x + Tan[c + d*x]/d

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\tan (c+d x)}{d}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -x+\frac {\tan (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.64 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\arctan (\tan (c+d x))}{d}+\frac {\tan (c+d x)}{d} \]

[In]

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

[Out]

-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(\frac {\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(21\)
default	\(\frac {\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(21\)
risch	\(-x +\frac {2 i}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\)	\(24\)
parallelrisch	\(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x d +d x -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(50\)
norman	\(\frac {x -\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(78\)

[In]

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

[Out]

1/d*(tan(d*x+c)-arctan(tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {d x \cos \left (d x + c\right ) - \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]

[In]

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

[Out]

-(d*x*cos(d*x + c) - sin(d*x + c))/(d*cos(d*x + c))

Sympy [F]

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

[In]

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

[Out]

Integral(sin(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. 64 vs. \(2 (14) = 28\).

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 4.57 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{{\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )\right )}}{d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {d x + c - \tan \left (d x + c\right )}{d} \]

[In]

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

[Out]

-(d*x + c - tan(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 23.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-x-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

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