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

Optimal. Leaf size=42 \[ \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2767, 2734} \[ \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a + a*Sin[x]),x]

[Out]

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

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2767

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(a + b*Sin[e + f*x])), x] - Dist[d/(a*b), Int[(c +
d*Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2
*n] || EqQ[c, 0])

Rubi steps

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

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Mathematica [B]  time = 0.09, size = 87, normalized size = 2.07 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (12 x \sin \left (\frac {x}{2}\right )-20 \sin \left (\frac {x}{2}\right )+3 \sin \left (\frac {3 x}{2}\right )-\sin \left (\frac {5 x}{2}\right )+4 (3 x+1) \cos \left (\frac {x}{2}\right )+3 \cos \left (\frac {3 x}{2}\right )+\cos \left (\frac {5 x}{2}\right )\right )}{8 a (\sin (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(a + a*Sin[x]),x]

[Out]

((Cos[x/2] + Sin[x/2])*(4*(1 + 3*x)*Cos[x/2] + 3*Cos[(3*x)/2] + Cos[(5*x)/2] - 20*Sin[x/2] + 12*x*Sin[x/2] + 3
*Sin[(3*x)/2] - Sin[(5*x)/2]))/(8*a*(1 + Sin[x]))

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fricas [A]  time = 0.47, size = 53, normalized size = 1.26 \[ \frac {\cos \relax (x)^{3} + 3 \, {\left (x + 1\right )} \cos \relax (x) + 2 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{2} - 3 \, x - \cos \relax (x) + 2\right )} \sin \relax (x) + 3 \, x + 2}{2 \, {\left (a \cos \relax (x) + a \sin \relax (x) + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.95, size = 56, normalized size = 1.33 \[ \frac {3 \, x}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 100, normalized size = 2.38 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {\tan \left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}+\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.65, size = 128, normalized size = 3.05 \[ \frac {\frac {\sin \relax (x)}{\cos \relax (x) + 1} + \frac {5 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 4}{a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {2 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {2 \, a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.91, size = 59, normalized size = 1.40 \[ \frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a + a*sin(x)),x)

[Out]

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

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sympy [B]  time = 2.79, size = 665, normalized size = 15.83 \[ \frac {3 x \tan ^{5}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 x \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x \tan {\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {10 \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {8}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*tan(x/2)**5/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) +
3*x*tan(x/2)**4/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a)
+ 6*x*tan(x/2)**3/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a)
+ 6*x*tan(x/2)**2/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a
) + 3*x*tan(x/2)/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a)
+ 3*x/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) + 6*tan(x/2
)**4/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) + 6*tan(x/2)
**3/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) + 10*tan(x/2)
**2/(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) + 2*tan(x/2)/
(2*a*tan(x/2)**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a) + 8/(2*a*tan(x/2)
**5 + 2*a*tan(x/2)**4 + 4*a*tan(x/2)**3 + 4*a*tan(x/2)**2 + 2*a*tan(x/2) + 2*a)

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