Integrand size = 13, antiderivative size = 66 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {7 x}{2 a^2}+\frac {16 \cos (x)}{3 a^2}-\frac {7 \cos (x) \sin (x)}{2 a^2}+\frac {8 \cos (x) \sin ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x) \sin ^3(x)}{3 (a+a \sin (x))^2} \]
7/2*x/a^2+16/3*cos(x)/a^2-7/2*cos(x)*sin(x)/a^2+8/3*cos(x)*sin(x)^2/a^2/(1 +sin(x))+1/3*cos(x)*sin(x)^3/(a+a*sin(x))^2
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (21 (-7+12 x) \cos \left (\frac {x}{2}\right )+(239-84 x) \cos \left (\frac {3 x}{2}\right )+3 \left (-5 \cos \left (\frac {5 x}{2}\right )+\cos \left (\frac {7 x}{2}\right )+2 (-50+56 x+(27+28 x) \cos (x)+6 \cos (2 x)+\cos (3 x)) \sin \left (\frac {x}{2}\right )\right )\right )}{48 a^2 (1+\sin (x))^2} \]
((Cos[x/2] + Sin[x/2])*(21*(-7 + 12*x)*Cos[x/2] + (239 - 84*x)*Cos[(3*x)/2 ] + 3*(-5*Cos[(5*x)/2] + Cos[(7*x)/2] + 2*(-50 + 56*x + (27 + 28*x)*Cos[x] + 6*Cos[2*x] + Cos[3*x])*Sin[x/2])))/(48*a^2*(1 + Sin[x])^2)
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3244, 3042, 3456, 3042, 3213}
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.
| \(\displaystyle \int \frac {\sin ^4(x)}{(a \sin (x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^4}{(a \sin (x)+a)^2}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\int \frac {\sin ^2(x) (3 a-5 a \sin (x))}{\sin (x) a+a}dx}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\int \frac {\sin (x)^2 (3 a-5 a \sin (x))}{\sin (x) a+a}dx}{3 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\frac {\int \sin (x) \left (16 a^2-21 a^2 \sin (x)\right )dx}{a^2}-\frac {8 \sin ^2(x) \cos (x)}{\sin (x)+1}}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\frac {\int \sin (x) \left (16 a^2-21 a^2 \sin (x)\right )dx}{a^2}-\frac {8 \sin ^2(x) \cos (x)}{\sin (x)+1}}{3 a^2}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\frac {-\frac {21 a^2 x}{2}-16 a^2 \cos (x)+\frac {21}{2} a^2 \sin (x) \cos (x)}{a^2}-\frac {8 \sin ^2(x) \cos (x)}{\sin (x)+1}}{3 a^2}\) |
(Cos[x]*Sin[x]^3)/(3*(a + a*Sin[x])^2) - ((-8*Cos[x]*Sin[x]^2)/(1 + Sin[x] ) + ((-21*a^2*x)/2 - 16*a^2*Cos[x] + (21*a^2*Cos[x]*Sin[x])/2)/a^2)/(3*a^2 )
3.1.12.3.1 Defintions of rubi rules used
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)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + 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] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Time = 0.54 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(\frac {8 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )-8 \left (\sec ^{3}\left (x \right )\right )+24 \cos \left (x \right )-44 \tan \left (x \right )+48 \sec \left (x \right )-3 \sin \left (2 x \right )+42 x +64}{12 a^{2}}\) | \(43\) |
| default | \(\frac {-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )+1}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {\tan \left (\frac {x}{2}\right )}{2}+2\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+7 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(80\) |
| risch | \(\frac {7 x}{2 a^{2}}+\frac {i {\mathrm e}^{2 i x}}{8 a^{2}}+\frac {{\mathrm e}^{i x}}{a^{2}}+\frac {{\mathrm e}^{-i x}}{a^{2}}-\frac {i {\mathrm e}^{-2 i x}}{8 a^{2}}+\frac {8 \,{\mathrm e}^{2 i x}-\frac {22}{3}+14 i {\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}\) | \(80\) |
| norman | \(\frac {\frac {7 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}+\frac {25 \tan \left (\frac {x}{2}\right )}{a}+\frac {92 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x}{2 a}+\frac {32}{3 a}+\frac {21 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {49 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {91 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {63 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {77 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {77 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {63 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {91 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {49 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {21 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {21 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}+\frac {108 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {84 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {161 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {130 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {314 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {140 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(273\) |
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=-\frac {3 \, \cos \left (x\right )^{4} - {\left (21 \, x - 31\right )} \cos \left (x\right )^{2} - 6 \, \cos \left (x\right )^{3} + {\left (21 \, x + 38\right )} \cos \left (x\right ) + {\left (3 \, \cos \left (x\right )^{3} + {\left (21 \, x + 40\right )} \cos \left (x\right ) + 9 \, \cos \left (x\right )^{2} + 42 \, x + 2\right )} \sin \left (x\right ) + 42 \, x - 2}{6 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \]
-1/6*(3*cos(x)^4 - (21*x - 31)*cos(x)^2 - 6*cos(x)^3 + (21*x + 38)*cos(x) + (3*cos(x)^3 + (21*x + 40)*cos(x) + 9*cos(x)^2 + 42*x + 2)*sin(x) + 42*x - 2)/(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. 1423 vs. \(2 (70) = 140\).
Time = 3.60 (sec) , antiderivative size = 1423, normalized size of antiderivative = 21.56 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\text {Too large to display} \]
21*x*tan(x/2)**7/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x /2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 63*x*tan(x/2)**6/(6*a**2*tan(x/2)**7 + 18*a* *2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x /2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 105*x*tan(x/2) **5/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a **2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan( x/2) + 6*a**2) + 147*x*tan(x/2)**4/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)* *6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30* a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 147*x*tan(x/2)**3/(6*a**2* tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2) **4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a** 2) + 105*x*tan(x/2)**2/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2 *tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2 )**2 + 18*a**2*tan(x/2) + 6*a**2) + 63*x*tan(x/2)/(6*a**2*tan(x/2)**7 + 18 *a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*ta n(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 21*x/(6*a** 2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/ 2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a **2) + 42*tan(x/2)**6/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a*...
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (56) = 112\).
Time = 0.31 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.00 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\frac {75 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {97 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {126 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {98 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 32}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {7 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {7 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {5 \, a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {3 \, a^{2} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} + \frac {7 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \]
1/3*(75*sin(x)/(cos(x) + 1) + 97*sin(x)^2/(cos(x) + 1)^2 + 126*sin(x)^3/(c os(x) + 1)^3 + 98*sin(x)^4/(cos(x) + 1)^4 + 63*sin(x)^5/(cos(x) + 1)^5 + 2 1*sin(x)^6/(cos(x) + 1)^6 + 32)/(a^2 + 3*a^2*sin(x)/(cos(x) + 1) + 5*a^2*s in(x)^2/(cos(x) + 1)^2 + 7*a^2*sin(x)^3/(cos(x) + 1)^3 + 7*a^2*sin(x)^4/(c os(x) + 1)^4 + 5*a^2*sin(x)^5/(cos(x) + 1)^5 + 3*a^2*sin(x)^6/(cos(x) + 1) ^6 + a^2*sin(x)^7/(cos(x) + 1)^7) + 7*arctan(sin(x)/(cos(x) + 1))/a^2
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {7 \, x}{2 \, a^{2}} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 4}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a^{2}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac {1}{2} \, x\right ) + 10\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
7/2*x/a^2 + (tan(1/2*x)^3 + 4*tan(1/2*x)^2 - tan(1/2*x) + 4)/((tan(1/2*x)^ 2 + 1)^2*a^2) + 2/3*(9*tan(1/2*x)^2 + 21*tan(1/2*x) + 10)/(a^2*(tan(1/2*x) + 1)^3)
Time = 6.47 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {7\,x}{2\,a^2}+\frac {7\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\frac {98\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{3}+42\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {97\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+25\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {32}{3}}{a^2\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]