\(\int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx\) [4]

Optimal result

Integrand size = 13, antiderivative size = 42 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \sin (x)} \]

[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))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2846, 2813} \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]

[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 2813

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 2846

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(42)=84\).

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.07 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (4 (1+3 x) \cos \left (\frac {x}{2}\right )+3 \cos \left (\frac {3 x}{2}\right )+\cos \left (\frac {5 x}{2}\right )-20 \sin \left (\frac {x}{2}\right )+12 x \sin \left (\frac {x}{2}\right )+3 \sin \left (\frac {3 x}{2}\right )-\sin \left (\frac {5 x}{2}\right )\right )}{8 a (1+\sin (x))} \]

[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]))

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(\frac {6 x +8-\sin \left (2 x \right )+4 \cos \left (x \right )-4 \tan \left (x \right )+4 \sec \left (x \right )}{4 a}\)	\(29\)
risch	\(\frac {3 x}{2 a}+\frac {{\mathrm e}^{i x}}{2 a}+\frac {{\mathrm e}^{-i x}}{2 a}+\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}-\frac {\sin \left (2 x \right )}{4 a}\)	\(52\)
default	\(\frac {\frac {16}{8 \tan \left (\frac {x}{2}\right )+8}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}+\tan ^{2}\left (\frac {x}{2}\right )-\frac {\tan \left (\frac {x}{2}\right )}{2}+1\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\)	\(58\)
norman	\(\frac {-\frac {\tan ^{5}\left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x}{2 a}+\frac {8}{3 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {9 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {4 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {\tan \left (\frac {x}{2}\right )}{3 a}+\frac {5 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(178\)

[In]

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

[Out]

1/4*(6*x+8-sin(2*x)+4*cos(x)-4*tan(x)+4*sec(x))/a

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\cos \left (x\right )^{3} + 3 \, {\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (39) = 78\).

Time = 0.88 (sec) , antiderivative size = 665, normalized size of antiderivative = 15.83 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\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} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (38) = 76\).

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.05 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 \, x}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.40 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

[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)
)