3.1.1 \(\int \frac {\sin ^4(x)}{a+a \cos (x)} \, dx\) [1]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {2761, 2715, 8} \begin {gather*} \frac {x}{2 a}-\frac {\sin ^3(x)}{3 a}-\frac {\sin (x) \cos (x)}{2 a} \end {gather*}

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 8

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

Rule 2715

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] && Integ
erQ[2*n]

Rule 2761

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]

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*}

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Mathematica [A]
time = 0.05, size = 25, normalized size = 0.81 \begin {gather*} \frac {6 x-3 \sin (x)-3 \sin (2 x)+\sin (3 x)}{12 a} \end {gather*}

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 [A]
time = 0.08, size = 48, normalized size = 1.55

method	result	size
risch	\(\frac {x}{2 a}-\frac {\sin \left (x \right )}{4 a}+\frac {\sin \left (3 x \right )}{12 a}-\frac {\sin \left (2 x \right )}{4 a}\)	\(33\)
default	\(\frac {\frac {16 \left (\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{16}-\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{6}-\frac {\tan \left (\frac {x}{2}\right )}{16}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\)	\(48\)
norman	\(\frac {\frac {\tan ^{7}\left (\frac {x}{2}\right )}{a}-\frac {\tan \left (\frac {x}{2}\right )}{a}-\frac {11 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {5 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {x}{2 a}+\frac {2 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/a*((1/16*tan(1/2*x)^5-1/6*tan(1/2*x)^3-1/16*tan(1/2*x))/(tan(1/2*x)^2+1)^3+1/16*arctan(tan(1/2*x)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (25) = 50\).
time = 0.49, size = 94, normalized size = 3.03 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 24, normalized size = 0.77 \begin {gather*} \frac {{\left (2 \, \cos \left (x\right )^{2} - 3 \, \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + 3 \, x}{6 \, a} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (22) = 44\).
time = 0.52, size = 294, normalized size = 9.48 \begin {gather*} \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 {gather*}

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 = 0.39, size = 45, normalized size = 1.45 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.38, size = 34, normalized size = 1.10 \begin {gather*} \frac {x}{2\,a}-\frac {\sin \left (x\right )}{3\,a}+\frac {{\cos \left (x\right )}^2\,\sin \left (x\right )}{3\,a}-\frac {\cos \left (x\right )\,\sin \left (x\right )}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a + a*cos(x)),x)

[Out]

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

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