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

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Rubi [A]  time = 0.048874, antiderivative size = 42, normalized size of antiderivative = 1., 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 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])

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]

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*}

Mathematica [B]  time = 0.0792168, 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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Maple [B]  time = 0.026, size = 100, normalized size = 2.4 \begin{align*}{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+3\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

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 = 2.5637, size = 173, normalized size = 4.12 \begin{align*} \frac{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \frac{3 \, \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)^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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Fricas [A]  time = 1.70878, size = 166, normalized size = 3.95 \begin{align*} \frac{\cos \left (x\right )^{3} + 3 \,{\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \,{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \end{align*}

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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Sympy [B]  time = 3.26524, size = 665, normalized size = 15.83 \begin{align*} \text{result too large to display} \end{align*}

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
)**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) - 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) - 2*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) - 4*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) + 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)

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Giac [A]  time = 1.80424, size = 76, normalized size = 1.81 \begin{align*} \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 )}} \end{align*}

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))