3.22 \(\int \frac{\sin ^3(x)}{(1+\cos (x))^3} \, dx\)

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

[Out]

2/(1 + Cos[x]) + Log[1 + Cos[x]]

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Rubi [A]  time = 0.0377091, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {2667, 43} \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(1 + Cos[x])^3,x]

[Out]

2/(1 + Cos[x]) + Log[1 + Cos[x]]

Rule 2667

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])

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0093198, size = 18, normalized size = 1.29 \[ \tan ^2\left (\frac{x}{2}\right )+2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

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 = 0.046, size = 15, normalized size = 1.1 \begin{align*} 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1}+\ln \left ( \cos \left ( x \right ) +1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.32383, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}

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 = 1.63419, size = 74, normalized size = 5.29 \begin{align*} \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{align*}

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]  time = 0.785423, size = 126, normalized size = 9. \begin{align*} \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{align*}

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 = 1.17238, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}

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)