Integrand size = 13, antiderivative size = 35 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^2}+\frac {5 \cos (x)}{3 a^2 (1+\sin (x))}-\frac {\cos (x)}{3 (a+a \sin (x))^2} \] Output:
x/a^2+5/3*cos(x)/a^2/(1+sin(x))-1/3*cos(x)/(a+a*sin(x))^2
Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (3 (-4+3 x) \cos \left (\frac {x}{2}\right )+(10-3 x) \cos \left (\frac {3 x}{2}\right )+6 (-3+2 x+x \cos (x)) \sin \left (\frac {x}{2}\right )\right )}{6 a^2 (1+\sin (x))^2} \] Input:
Integrate[Sin[x]^2/(a + a*Sin[x])^2,x]
Output:
((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)
Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 3237, 25, 3042, 3214, 3042, 3127}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin ^2(x)}{(a \sin (x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)^2}{(a \sin (x)+a)^2}dx\) |
\(\Big \downarrow \) 3237 |
\(\displaystyle \frac {\int -\frac {2 a-3 a \sin (x)}{\sin (x) a+a}dx}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {2 a-3 a \sin (x)}{\sin (x) a+a}dx}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {2 a-3 a \sin (x)}{\sin (x) a+a}dx}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle -\frac {5 a \int \frac {1}{\sin (x) a+a}dx-3 x}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 a \int \frac {1}{\sin (x) a+a}dx-3 x}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -\frac {-\frac {5 a \cos (x)}{a \sin (x)+a}-3 x}{3 a^2}-\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
Input:
Int[Sin[x]^2/(a + a*Sin[x])^2,x]
Output:
-1/3*Cos[x]/(a + a*Sin[x])^2 - (-3*x - (5*a*Cos[x])/(a + a*Sin[x]))/(3*a^2 )
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(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]
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] - Simp[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)]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11
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 | \(\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{4 \tan \left (\frac {x}{2}\right )+4}}{a^{2}}\) | \(44\) |
parallelrisch | \(\frac {3 \tan \left (\frac {x}{2}\right )^{3} x +\left (9 x +6\right ) \tan \left (\frac {x}{2}\right )^{2}+\left (9 x +18\right ) \tan \left (\frac {x}{2}\right )+3 x +8}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(50\) |
norman | \(\frac {\frac {x}{a}+\frac {x \tan \left (\frac {x}{2}\right )^{7}}{a}+\frac {2 \tan \left (\frac {x}{2}\right )^{6}}{a}+\frac {6 \tan \left (\frac {x}{2}\right )^{5}}{a}+\frac {12 \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {8}{3 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{a}+\frac {5 x \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {7 x \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {7 x \tan \left (\frac {x}{2}\right )^{4}}{a}+\frac {5 x \tan \left (\frac {x}{2}\right )^{5}}{a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{6}}{a}+\frac {6 \tan \left (\frac {x}{2}\right )}{a}+\frac {22 \tan \left (\frac {x}{2}\right )^{2}}{3 a}+\frac {20 \tan \left (\frac {x}{2}\right )^{4}}{3 a}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(179\) |
Input:
int(sin(x)^2/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
x/a^2+2/3*(6*exp(2*I*x)-5+9*I*exp(I*x))/(exp(I*x)+I)^3/a^2
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (31) = 62\).
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.34 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {{\left (3 \, x - 5\right )} \cos \left (x\right )^{2} - {\left (3 \, x + 4\right )} \cos \left (x\right ) - {\left ({\left (3 \, x + 5\right )} \cos \left (x\right ) + 6 \, x + 1\right )} \sin \left (x\right ) - 6 \, x + 1}{3 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(sin(x)^2/(a+a*sin(x))^2,x, algorithm="fricas")
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (32) = 64\).
Time = 1.05 (sec) , antiderivative size = 321, normalized size of antiderivative = 9.17 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {3 x \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {3 x}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {6 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {18 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {8}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \] Input:
integrate(sin(x)**2/(a+a*sin(x))**2,x)
Output:
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*tan(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)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (31) = 62\).
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.57 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 4\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \] Input:
integrate(sin(x)^2/(a+a*sin(x))^2,x, algorithm="maxima")
Output:
2/3*(9*sin(x)/(cos(x) + 1) + 3*sin(x)^2/(cos(x) + 1)^2 + 4)/(a^2 + 3*a^2*s in(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
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 4\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \] Input:
integrate(sin(x)^2/(a+a*sin(x))^2,x, algorithm="giac")
Output:
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)
Time = 17.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \] Input:
int(sin(x)^2/(a + a*sin(x))^2,x)
Output:
x/a^2 + (6*tan(x/2) + 2*tan(x/2)^2 + 8/3)/(a^2*(tan(x/2) + 1)^3)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.09 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {3 \tan \left (\frac {x}{2}\right )^{3} x -2 \tan \left (\frac {x}{2}\right )^{3}+9 \tan \left (\frac {x}{2}\right )^{2} x +9 \tan \left (\frac {x}{2}\right ) x +12 \tan \left (\frac {x}{2}\right )+3 x +6}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )^{3}+3 \tan \left (\frac {x}{2}\right )^{2}+3 \tan \left (\frac {x}{2}\right )+1\right )} \] Input:
int(sin(x)^2/(a+a*sin(x))^2,x)
Output:
(3*tan(x/2)**3*x - 2*tan(x/2)**3 + 9*tan(x/2)**2*x + 9*tan(x/2)*x + 12*tan (x/2) + 3*x + 6)/(3*a**2*(tan(x/2)**3 + 3*tan(x/2)**2 + 3*tan(x/2) + 1))