3.1.5 \(\int \frac {\sin ^2(x)}{a+a \sin (x)} \, dx\) [5]

Optimal. Leaf size=27 \[ -\frac {x}{a}-\frac {\cos (x)}{a}-\frac {\cos (x)}{a (1+\sin (x))} \]

[Out]

-x/a-cos(x)/a-cos(x)/a/(1+sin(x))

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Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2825, 12, 2814, 2727} \begin {gather*} -\frac {x}{a}-\frac {\cos (x)}{a}-\frac {\cos (x)}{a (\sin (x)+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + a*Sin[x]),x]

[Out]

-(x/a) - Cos[x]/a - Cos[x]/(a*(1 + Sin[x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2814

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

Rule 2825

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 48, normalized size = 1.78 \begin {gather*} -\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right ) (x+\cos (x))+(-2+x+\cos (x)) \sin \left (\frac {x}{2}\right )\right )}{a (1+\sin (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + a*Sin[x]),x]

[Out]

-(((Cos[x/2] + Sin[x/2])*(Cos[x/2]*(x + Cos[x]) + (-2 + x + Cos[x])*Sin[x/2]))/(a*(1 + Sin[x])))

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Maple [A]
time = 0.09, size = 36, normalized size = 1.33

method	result	size
default	\(\frac {-\frac {2}{\tan \left (\frac {x}{2}\right )+1}-\frac {2}{\tan ^{2}\left (\frac {x}{2}\right )+1}-2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\)	\(36\)
risch	\(-\frac {x}{a}-\frac {{\mathrm e}^{i x}}{2 a}-\frac {{\mathrm e}^{-i x}}{2 a}-\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}\)	\(43\)
norman	\(\frac {-\frac {4}{a}-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {x}{a}-\frac {x \tan \left (\frac {x}{2}\right )}{a}-\frac {2 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a+a*sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (27) = 54\).
time = 0.52, size = 78, normalized size = 2.89 \begin {gather*} -\frac {2 \, {\left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right )}}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}} - \frac {2 \, \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)^2/(a+a*sin(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 35, normalized size = 1.30 \begin {gather*} -\frac {{\left (x + 2\right )} \cos \left (x\right ) + \cos \left (x\right )^{2} + {\left (x + \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + x + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(a+a*sin(x)),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (19) = 38\).
time = 0.54, size = 221, normalized size = 8.19 \begin {gather*} - \frac {x \tan ^{3}{\left (\frac {x}{2} \right )}}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {x \tan ^{2}{\left (\frac {x}{2} \right )}}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {x \tan {\left (\frac {x}{2} \right )}}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {x}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {2 \tan ^{2}{\left (\frac {x}{2} \right )}}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} - \frac {4}{a \tan ^{3}{\left (\frac {x}{2} \right )} + a \tan ^{2}{\left (\frac {x}{2} \right )} + a \tan {\left (\frac {x}{2} \right )} + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(a+a*sin(x)),x)

[Out]

-x*tan(x/2)**3/(a*tan(x/2)**3 + a*tan(x/2)**2 + a*tan(x/2) + a) - x*tan(x/2)**2/(a*tan(x/2)**3 + a*tan(x/2)**2
+ a*tan(x/2) + a) - x*tan(x/2)/(a*tan(x/2)**3 + a*tan(x/2)**2 + a*tan(x/2) + a) - x/(a*tan(x/2)**3 + a*tan(x/
2)**2 + a*tan(x/2) + a) - 2*tan(x/2)**2/(a*tan(x/2)**3 + a*tan(x/2)**2 + a*tan(x/2) + a) - 2*tan(x/2)/(a*tan(x
/2)**3 + a*tan(x/2)**2 + a*tan(x/2) + a) - 4/(a*tan(x/2)**3 + a*tan(x/2)**2 + a*tan(x/2) + a)

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Giac [A]
time = 0.56, size = 44, normalized size = 1.63 \begin {gather*} -\frac {x}{a} - \frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(a+a*sin(x)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 6.77, size = 46, normalized size = 1.70 \begin {gather*} -\frac {x}{a}-\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a + a*sin(x)),x)

[Out]

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

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