3.1.11 \(\int \frac {\sin (2 x)}{1-\cos (2 x)} \, dx\) [11]

Optimal. Leaf size=3 \[ \log (\sin (x)) \]

[Out]

ln(sin(x))

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(13\) vs. \(2(3)=6\).
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 4.33, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2746, 31} \begin {gather*} \frac {1}{2} \log (1-\cos (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[2*x]/(1 - Cos[2*x]),x]

[Out]

Log[1 - Cos[2*x]]/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\sin (2 x)}{1-\cos (2 x)} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\cos (2 x)\right )\\ &=\frac {1}{2} \log (1-\cos (2 x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 3, normalized size = 1.00 \begin {gather*} \log (\sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[2*x]/(1 - Cos[2*x]),x]

[Out]

Log[Sin[x]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11\) vs. \(2(3)=6\).
time = 0.04, size = 12, normalized size = 4.00

method	result	size
derivativedivides	\(\frac {\ln \left (1-\cos \left (2 x \right )\right )}{2}\)	\(12\)
default	\(\frac {\ln \left (1-\cos \left (2 x \right )\right )}{2}\)	\(12\)
norman	\(-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}+\ln \left (\tan \left (x \right )\right )\)	\(14\)
risch	\(-i x +\ln \left ({\mathrm e}^{2 i x}-1\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(1-cos(2*x)),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(1-cos(2*x))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 9 vs. \(2 (3) = 6\).
time = 0.27, size = 9, normalized size = 3.00 \begin {gather*} \frac {1}{2} \, \log \left (\cos \left (2 \, x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(1-cos(2*x)),x, algorithm="maxima")

[Out]

1/2*log(cos(2*x) - 1)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (3) = 6\).
time = 0.37, size = 11, normalized size = 3.67 \begin {gather*} \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(1-cos(2*x)),x, algorithm="fricas")

[Out]

1/2*log(-1/2*cos(2*x) + 1/2)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 8 vs. \(2 (3) = 6\).
time = 0.04, size = 8, normalized size = 2.67 \begin {gather*} \frac {\log {\left (\cos {\left (2 x \right )} - 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(1-cos(2*x)),x)

[Out]

log(cos(2*x) - 1)/2

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (3) = 6\).
time = 0.42, size = 11, normalized size = 3.67 \begin {gather*} \frac {1}{2} \, \log \left (-\cos \left (2 \, x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(1-cos(2*x)),x, algorithm="giac")

[Out]

1/2*log(-cos(2*x) + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 9, normalized size = 3.00 \begin {gather*} \frac {\ln \left (-{\sin \left (x\right )}^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sin(2*x)/(cos(2*x) - 1),x)

[Out]

log(-sin(x)^2)/2

________________________________________________________________________________________