3.90 \(\int \csc (2 x) \sin (x) \, dx\)

Optimal. Leaf size=7 \[ \frac{1}{2} \tanh ^{-1}(\sin (x)) \]

[Out]

ArcTanh[Sin[x]]/2

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Rubi [A]  time = 0.0119241, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4288, 3770} \[ \frac{1}{2} \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[2*x]*Sin[x],x]

[Out]

ArcTanh[Sin[x]]/2

Rule 4288

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc (2 x) \sin (x) \, dx &=\frac{1}{2} \int \sec (x) \, dx\\ &=\frac{1}{2} \tanh ^{-1}(\sin (x))\\ \end{align*}

Mathematica [B]  time = 0.0062136, size = 37, normalized size = 5.29 \[ \frac{1}{2} \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*x]*Sin[x],x]

[Out]

(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/2

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Maple [A]  time = 0.018, size = 9, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*x)*sin(x),x)

[Out]

1/2*ln(sec(x)+tan(x))

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Maxima [B]  time = 1.54641, size = 47, normalized size = 6.71 \begin{align*} \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*sin(x),x, algorithm="maxima")

[Out]

1/4*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) - 1/4*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)

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Fricas [B]  time = 2.3467, size = 59, normalized size = 8.43 \begin{align*} \frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*sin(x),x, algorithm="fricas")

[Out]

1/4*log(sin(x) + 1) - 1/4*log(-sin(x) + 1)

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Sympy [B]  time = 6.40099, size = 15, normalized size = 2.14 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*sin(x),x)

[Out]

-log(sin(x) - 1)/4 + log(sin(x) + 1)/4

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Giac [B]  time = 1.13954, size = 34, normalized size = 4.86 \begin{align*} \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x\right )} + \sin \left (x\right ) + 2 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x\right )} + \sin \left (x\right ) - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*sin(x),x, algorithm="giac")

[Out]

1/8*log(abs(1/sin(x) + sin(x) + 2)) - 1/8*log(abs(1/sin(x) + sin(x) - 2))