3.1.22 \(\int \frac {\sin ^3(x)}{(1+\cos (x))^3} \, dx\) [22]

Optimal. Leaf size=14 \[ \frac {2}{1+\cos (x)}+\log (1+\cos (x)) \]

[Out]

2/(cos(x)+1)+ln(cos(x)+1)

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2746, 45} \begin {gather*} \frac {2}{\cos (x)+1}+\log (\cos (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(1 + Cos[x])^3,x]

[Out]

2/(1 + Cos[x]) + Log[1 + Cos[x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 ^3(x)}{(1+\cos (x))^3} \, dx &=-\text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {2}{1+\cos (x)}+\log (1+\cos (x))\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.29 \begin {gather*} 2 \log \left (\cos \left (\frac {x}{2}\right )\right )+\tan ^2\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(1 + Cos[x])^3,x]

[Out]

2*Log[Cos[x/2]] + Tan[x/2]^2

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Maple [A]
time = 0.07, size = 15, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {2}{\cos \left (x \right )+1}+\ln \left (\cos \left (x \right )+1\right )\)	\(15\)
default	\(\frac {2}{\cos \left (x \right )+1}+\ln \left (\cos \left (x \right )+1\right )\)	\(15\)
risch	\(-i x +\frac {4 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+1\right )^{2}}+2 \ln \left ({\mathrm e}^{i x}+1\right )\)	\(32\)
norman	\(\frac {\tan ^{8}\left (\frac {x}{2}\right )-2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )-1}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(cos(x)+1)^3,x,method=_RETURNVERBOSE)

[Out]

2/(cos(x)+1)+ln(cos(x)+1)

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Maxima [A]
time = 0.26, size = 14, normalized size = 1.00 \begin {gather*} \frac {2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(1+cos(x))^3,x, algorithm="maxima")

[Out]

2/(cos(x) + 1) + log(cos(x) + 1)

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Fricas [A]
time = 0.36, size = 21, normalized size = 1.50 \begin {gather*} \frac {{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2}{\cos \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(1+cos(x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (12) = 24\).
time = 0.27, size = 126, normalized size = 9.00 \begin {gather*} \frac {2 \log {\left (\cos {\left (x \right )} + 1 \right )} \cos ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} + \frac {4 \log {\left (\cos {\left (x \right )} + 1 \right )} \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} + \frac {2 \log {\left (\cos {\left (x \right )} + 1 \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} + \frac {\sin ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} + \frac {2 \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} + \frac {2}{2 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/(1+cos(x))**3,x)

[Out]

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

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Giac [A]
time = 0.44, size = 14, normalized size = 1.00 \begin {gather*} \frac {2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(1+cos(x))^3,x, algorithm="giac")

[Out]

2/(cos(x) + 1) + log(cos(x) + 1)

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Mupad [B]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \ln \left (\cos \left (x\right )+1\right )+\frac {2}{\cos \left (x\right )+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(cos(x) + 1)^3,x)

[Out]

log(cos(x) + 1) + 2/(cos(x) + 1)

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