3.1 \(\int \frac{\sin ^4(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{x}{2 a}-\frac{\sin ^3(x)}{3 a}-\frac{\sin (x) \cos (x)}{2 a} \]

[Out]

x/(2*a) - (Cos[x]*Sin[x])/(2*a) - Sin[x]^3/(3*a)

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Rubi [A]  time = 0.0428443, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ \frac{x}{2 a}-\frac{\sin ^3(x)}{3 a}-\frac{\sin (x) \cos (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + a*Cos[x]),x]

[Out]

x/(2*a) - (Cos[x]*Sin[x])/(2*a) - Sin[x]^3/(3*a)

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sin ^4(x)}{a+a \cos (x)} \, dx &=-\frac{\sin ^3(x)}{3 a}+\frac{\int \sin ^2(x) \, dx}{a}\\ &=-\frac{\cos (x) \sin (x)}{2 a}-\frac{\sin ^3(x)}{3 a}+\frac{\int 1 \, dx}{2 a}\\ &=\frac{x}{2 a}-\frac{\cos (x) \sin (x)}{2 a}-\frac{\sin ^3(x)}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0399624, size = 25, normalized size = 0.81 \[ \frac{6 x-3 \sin (x)-3 \sin (2 x)+\sin (3 x)}{12 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + a*Cos[x]),x]

[Out]

(6*x - 3*Sin[x] - 3*Sin[2*x] + Sin[3*x])/(12*a)

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Maple [B]  time = 0.046, size = 68, normalized size = 2.2 \begin{align*}{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-{\frac{8}{3\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}+{\frac{x}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a+a*cos(x)),x)

[Out]

1/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)^5-8/3/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)^3-1/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)+1/
2*x/a

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Maxima [B]  time = 1.76684, size = 127, normalized size = 4.1 \begin{align*} -\frac{\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{8 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{3 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{3 \,{\left (a + \frac{3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} + \frac{\arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*cos(x)),x, algorithm="maxima")

[Out]

-1/3*(3*sin(x)/(cos(x) + 1) + 8*sin(x)^3/(cos(x) + 1)^3 - 3*sin(x)^5/(cos(x) + 1)^5)/(a + 3*a*sin(x)^2/(cos(x)
+ 1)^2 + 3*a*sin(x)^4/(cos(x) + 1)^4 + a*sin(x)^6/(cos(x) + 1)^6) + arctan(sin(x)/(cos(x) + 1))/a

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Fricas [A]  time = 1.58075, size = 68, normalized size = 2.19 \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{2} - 3 \, \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + 3 \, x}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*cos(x)),x, algorithm="fricas")

[Out]

1/6*((2*cos(x)^2 - 3*cos(x) - 2)*sin(x) + 3*x)/a

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Sympy [B]  time = 2.06742, size = 294, normalized size = 9.48 \begin{align*} \frac{3 x \tan ^{6}{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} + \frac{9 x \tan ^{4}{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} + \frac{9 x \tan ^{2}{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} + \frac{3 x}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} + \frac{6 \tan ^{5}{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} - \frac{16 \tan ^{3}{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} - \frac{6 \tan{\left (\frac{x}{2} \right )}}{6 a \tan ^{6}{\left (\frac{x}{2} \right )} + 18 a \tan ^{4}{\left (\frac{x}{2} \right )} + 18 a \tan ^{2}{\left (\frac{x}{2} \right )} + 6 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**4/(a+a*cos(x)),x)

[Out]

3*x*tan(x/2)**6/(6*a*tan(x/2)**6 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**2 + 6*a) + 9*x*tan(x/2)**4/(6*a*tan(x/2)*
*6 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**2 + 6*a) + 9*x*tan(x/2)**2/(6*a*tan(x/2)**6 + 18*a*tan(x/2)**4 + 18*a*t
an(x/2)**2 + 6*a) + 3*x/(6*a*tan(x/2)**6 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**2 + 6*a) + 6*tan(x/2)**5/(6*a*tan
(x/2)**6 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**2 + 6*a) - 16*tan(x/2)**3/(6*a*tan(x/2)**6 + 18*a*tan(x/2)**4 + 1
8*a*tan(x/2)**2 + 6*a) - 6*tan(x/2)/(6*a*tan(x/2)**6 + 18*a*tan(x/2)**4 + 18*a*tan(x/2)**2 + 6*a)

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Giac [A]  time = 1.10023, size = 61, normalized size = 1.97 \begin{align*} \frac{x}{2 \, a} + \frac{3 \, \tan \left (\frac{1}{2} \, x\right )^{5} - 8 \, \tan \left (\frac{1}{2} \, x\right )^{3} - 3 \, \tan \left (\frac{1}{2} \, x\right )}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*cos(x)),x, algorithm="giac")

[Out]

1/2*x/a + 1/3*(3*tan(1/2*x)^5 - 8*tan(1/2*x)^3 - 3*tan(1/2*x))/((tan(1/2*x)^2 + 1)^3*a)