3.2 \(\int \frac{\sin ^3(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a} \]

[Out]

-(Cos[x]/a) + Cos[x]^2/(2*a)

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Rubi [A]  time = 0.0399363, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a + a*Cos[x]),x]

[Out]

-(Cos[x]/a) + Cos[x]^2/(2*a)

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])

Rubi steps

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

Mathematica [A]  time = 0.0120588, size = 13, normalized size = 0.68 \[ \frac{2 \sin ^4\left (\frac{x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(a + a*Cos[x]),x]

[Out]

(2*Sin[x/2]^4)/a

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Maple [A]  time = 0.039, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}-\cos \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a+a*cos(x)),x)

[Out]

1/a*(1/2*cos(x)^2-cos(x))

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Maxima [A]  time = 1.0833, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(x)^2 - 2*cos(x))/a

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Fricas [A]  time = 1.54458, size = 39, normalized size = 2.05 \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(x)^2 - 2*cos(x))/a

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Sympy [B]  time = 0.976852, size = 88, normalized size = 4.63 \begin{align*} \frac{4 \tan ^{4}{\left (\frac{x}{2} \right )}}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{4 \tan ^{2}{\left (\frac{x}{2} \right )}}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{2}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*tan(x/2)**4/(3*a*tan(x/2)**4 + 6*a*tan(x/2)**2 + 3*a) - 4*tan(x/2)**2/(3*a*tan(x/2)**4 + 6*a*tan(x/2)**2 + 3
*a) - 2/(3*a*tan(x/2)**4 + 6*a*tan(x/2)**2 + 3*a)

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Giac [A]  time = 1.14154, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(x)^2 - 2*cos(x))/a