3.1.1 \(\int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx\) [1]

Optimal. Leaf size=33 \[ \frac {3 x}{8 a}-\frac {3 \cos (x) \sin (x)}{8 a}-\frac {\cos (x) \sin ^3(x)}{4 a} \]

[Out]

3/8*x/a-3/8*cos(x)*sin(x)/a-1/4*cos(x)*sin(x)^3/a

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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3254, 2715, 8} \begin {gather*} \frac {3 x}{8 a}-\frac {\sin ^3(x) \cos (x)}{4 a}-\frac {3 \sin (x) \cos (x)}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^6/(a - a*Cos[x]^2),x]

[Out]

(3*x)/(8*a) - (3*Cos[x]*Sin[x])/(8*a) - (Cos[x]*Sin[x]^3)/(4*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 3254

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx &=\frac {\int \sin ^4(x) \, dx}{a}\\ &=-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {3 \int \sin ^2(x) \, dx}{4 a}\\ &=-\frac {3 \cos (x) \sin (x)}{8 a}-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {3 \int 1 \, dx}{8 a}\\ &=\frac {3 x}{8 a}-\frac {3 \cos (x) \sin (x)}{8 a}-\frac {\cos (x) \sin ^3(x)}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.79 \begin {gather*} \frac {\frac {3 x}{8}-\frac {1}{4} \sin (2 x)+\frac {1}{32} \sin (4 x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^6/(a - a*Cos[x]^2),x]

[Out]

((3*x)/8 - Sin[2*x]/4 + Sin[4*x]/32)/a

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Maple [A]
time = 0.09, size = 31, normalized size = 0.94

method	result	size
risch	\(\frac {3 x}{8 a}+\frac {\sin \left (4 x \right )}{32 a}-\frac {\sin \left (2 x \right )}{4 a}\)	\(26\)
default	\(\frac {\frac {-\frac {5 \left (\tan ^{3}\left (x \right )\right )}{8}-\frac {3 \tan \left (x \right )}{8}}{\left (1+\tan ^{2}\left (x \right )\right )^{2}}+\frac {3 \arctan \left (\tan \left (x \right )\right )}{8}}{a}\)	\(31\)
norman	\(\frac {-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {17 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {7 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {7 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {17 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {3 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{8 a}+\frac {9 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {45 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {45 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {9 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {3 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{8 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6} \tan \left (\frac {x}{2}\right )}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^6/(a-a*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 37, normalized size = 1.12 \begin {gather*} -\frac {5 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{8 \, {\left (a \tan \left (x\right )^{4} + 2 \, a \tan \left (x\right )^{2} + a\right )}} + \frac {3 \, x}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="maxima")

[Out]

-1/8*(5*tan(x)^3 + 3*tan(x))/(a*tan(x)^4 + 2*a*tan(x)^2 + a) + 3/8*x/a

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Fricas [A]
time = 0.38, size = 23, normalized size = 0.70 \begin {gather*} \frac {{\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (29) = 58\).
time = 2.26, size = 473, normalized size = 14.33 \begin {gather*} \frac {3 x \tan ^{8}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{6}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {18 x \tan ^{4}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{2}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {3 x}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {6 \tan ^{7}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {22 \tan ^{5}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {22 \tan ^{3}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {6 \tan {\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**6/(a-a*cos(x)**2),x)

[Out]

3*x*tan(x/2)**8/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/
2)**6/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 18*x*tan(x/2)**4/(8*a
*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/2)**2/(8*a*tan(x/2)*
*8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 3*x/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 +
48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 6*tan(x/2)**7/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2
)**4 + 32*a*tan(x/2)**2 + 8*a) + 22*tan(x/2)**5/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*
tan(x/2)**2 + 8*a) - 22*tan(x/2)**3/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2
+ 8*a) - 6*tan(x/2)/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a)

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Giac [A]
time = 0.41, size = 31, normalized size = 0.94 \begin {gather*} \frac {3 \, x}{8 \, a} - \frac {5 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{8 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{2} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.11, size = 25, normalized size = 0.76 \begin {gather*} \frac {\sin \left (4\,x\right )}{32\,a}-\frac {\sin \left (2\,x\right )}{4\,a}+\frac {3\,x}{8\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^6/(a - a*cos(x)^2),x)

[Out]

sin(4*x)/(32*a) - sin(2*x)/(4*a) + (3*x)/(8*a)

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