3.1.14 \(\int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx\) [14]

Optimal. Leaf size=35 \[ \frac {x}{a^2}+\frac {5 \cos (x)}{3 a^2 (1+\sin (x))}-\frac {\cos (x)}{3 (a+a \sin (x))^2} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2837, 2814, 2727} \begin {gather*} \frac {x}{a^2}+\frac {5 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {\cos (x)}{3 (a \sin (x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + a*Sin[x])^2,x]

[Out]

x/a^2 + (5*Cos[x])/(3*a^2*(1 + Sin[x])) - Cos[x]/(3*(a + a*Sin[x])^2)

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 2837

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

Rubi steps

\begin {align*} \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx &=-\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {-2 a+3 a \sin (x)}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {x}{a^2}-\frac {\cos (x)}{3 (a+a \sin (x))^2}-\frac {5 \int \frac {1}{a+a \sin (x)} \, dx}{3 a}\\ &=\frac {x}{a^2}-\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {5 \cos (x)}{3 \left (a^2+a^2 \sin (x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 69, normalized size = 1.97 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (3 (-4+3 x) \cos \left (\frac {x}{2}\right )+(10-3 x) \cos \left (\frac {3 x}{2}\right )+6 (-3+2 x+x \cos (x)) \sin \left (\frac {x}{2}\right )\right )}{6 a^2 (1+\sin (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + a*Sin[x])^2,x]

[Out]

((Cos[x/2] + Sin[x/2])*(3*(-4 + 3*x)*Cos[x/2] + (10 - 3*x)*Cos[(3*x)/2] + 6*(-3 + 2*x + x*Cos[x])*Sin[x/2]))/(
6*a^2*(1 + Sin[x])^2)

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Maple [A]
time = 0.10, size = 44, normalized size = 1.26

method	result	size
risch	\(\frac {x}{a^{2}}+\frac {4 \,{\mathrm e}^{2 i x}-\frac {10}{3}+6 i {\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}\)	\(39\)
default	\(\frac {-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{4 \tan \left (\frac {x}{2}\right )+4}+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\)	\(44\)
norman	\(\frac {\frac {x}{a}+\frac {x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {12 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {8}{3 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{a}+\frac {5 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \tan \left (\frac {x}{2}\right )}{a}+\frac {22 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {20 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/a^2*(-1/6/(tan(1/2*x)+1)^3+1/4/(tan(1/2*x)+1)^2+1/4/(tan(1/2*x)+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. 90 vs. \(2 (31) = 62\).
time = 0.51, size = 90, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (\frac {9 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 4\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(9*sin(x)/(cos(x) + 1) + 3*sin(x)^2/(cos(x) + 1)^2 + 4)/(a^2 + 3*a^2*sin(x)/(cos(x) + 1) + 3*a^2*sin(x)^2/
(cos(x) + 1)^2 + a^2*sin(x)^3/(cos(x) + 1)^3) + 2*arctan(sin(x)/(cos(x) + 1))/a^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (31) = 62\).
time = 0.34, size = 82, normalized size = 2.34 \begin {gather*} \frac {{\left (3 \, x - 5\right )} \cos \left (x\right )^{2} - {\left (3 \, x + 4\right )} \cos \left (x\right ) - {\left ({\left (3 \, x + 5\right )} \cos \left (x\right ) + 6 \, x + 1\right )} \sin \left (x\right ) - 6 \, x + 1}{3 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((3*x - 5)*cos(x)^2 - (3*x + 4)*cos(x) - ((3*x + 5)*cos(x) + 6*x + 1)*sin(x) - 6*x + 1)/(a^2*cos(x)^2 - a^
2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (32) = 64\).
time = 1.35, size = 321, normalized size = 9.17 \begin {gather*} \frac {3 x \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {3 x}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {6 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {18 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {8}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*tan(x/2)**3/(3*a**2*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) + 9*x*tan(x/2)**2/(3*a**2
*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) + 9*x*tan(x/2)/(3*a**2*tan(x/2)**3 + 9*a**2*tan(
x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) + 3*x/(3*a**2*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2)
+ 6*tan(x/2)**2/(3*a**2*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) + 18*tan(x/2)/(3*a**2*ta
n(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a**2*tan(x/2) + 3*a**2) + 8/(3*a**2*tan(x/2)**3 + 9*a**2*tan(x/2)**2 + 9*a*
*2*tan(x/2) + 3*a**2)

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Giac [A]
time = 0.50, size = 35, normalized size = 1.00 \begin {gather*} \frac {x}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 4\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^2 + 2/3*(3*tan(1/2*x)^2 + 9*tan(1/2*x) + 4)/(a^2*(tan(1/2*x) + 1)^3)

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Mupad [B]
time = 6.48, size = 34, normalized size = 0.97 \begin {gather*} \frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a + a*sin(x))^2,x)

[Out]

x/a^2 + (6*tan(x/2) + 2*tan(x/2)^2 + 8/3)/(a^2*(tan(x/2) + 1)^3)

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