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

Optimal. Leaf size=14 \[ \frac{\tan (c+d x)}{d}-x \]

[Out]

-x + Tan[c + d*x]/d

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Rubi [A]  time = 0.146682, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {321, 203} \[ \frac{\tan (c+d x)}{d}-x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x + Tan[c + d*x]/d

Rule 321

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]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan (c+d x)}{d}-\frac{\operatorname{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]  time = 0.0079127, size = 23, normalized size = 1.64 \[ \frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.079, size = 24, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.62272, size = 86, normalized size = 6.14 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 0.470194, size = 72, normalized size = 5.14 \begin{align*} -\frac{d x \cos \left (d x + c\right ) - \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(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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\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(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)

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Giac [A]  time = 1.12026, size = 24, normalized size = 1.71 \begin{align*} -\frac{d x + c - \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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