3.306 \(\int (\csc (x)-\sin (x)) \, dx\)

Optimal. Leaf size=8 \[ \cos (x)-\tanh ^{-1}(\cos (x)) \]

[Out]

-arctanh(cos(x))+cos(x)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3770, 2638} \[ \cos (x)-\tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x] - Sin[x],x]

[Out]

-ArcTanh[Cos[x]] + Cos[x]

Rule 2638

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

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 (x)-\sin (x)) \, dx &=\int \csc (x) \, dx-\int \sin (x) \, dx\\ &=-\tanh ^{-1}(\cos (x))+\cos (x)\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 19, normalized size = 2.38 \[ \cos (x)+\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x] - Sin[x],x]

[Out]

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

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fricas [B]  time = 0.94, size = 21, normalized size = 2.62 \[ \cos \relax (x) - \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)-sin(x),x, algorithm="fricas")

[Out]

cos(x) - 1/2*log(1/2*cos(x) + 1/2) + 1/2*log(-1/2*cos(x) + 1/2)

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giac [A]  time = 0.15, size = 9, normalized size = 1.12 \[ \cos \relax (x) + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)-sin(x),x, algorithm="giac")

[Out]

cos(x) + log(abs(tan(1/2*x)))

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maple [A]  time = 0.00, size = 12, normalized size = 1.50 \[ \cos \relax (x )-\ln \left (\cot \relax (x )+\csc \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)-sin(x),x)

[Out]

cos(x)-ln(cot(x)+csc(x))

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maxima [A]  time = 0.32, size = 11, normalized size = 1.38 \[ \cos \relax (x) - \log \left (\cot \relax (x) + \csc \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)-sin(x),x, algorithm="maxima")

[Out]

cos(x) - log(cot(x) + csc(x))

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mupad [B]  time = 0.02, size = 8, normalized size = 1.00 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+\cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x) - sin(x),x)

[Out]

log(tan(x/2)) + cos(x)

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sympy [B]  time = 0.09, size = 19, normalized size = 2.38 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} + \cos {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)-sin(x),x)

[Out]

log(cos(x) - 1)/2 - log(cos(x) + 1)/2 + cos(x)

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