Integrand size = 29, antiderivative size = 83 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 x}{a^2}+\frac {3 \cos (c+d x)}{a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))} \]
3*x/a^2+3*cos(d*x+c)/a^2/d-1/3*cos(d*x+c)^3/a^2/d-cos(d*x+c)*sin(d*x+c)/a^ 2/d+2*cos(d*x+c)/a^2/d/(1+sin(d*x+c))
Time = 1.01 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {-72 d x \cos \left (\frac {d x}{2}\right )-31 \cos \left (c+\frac {d x}{2}\right )-27 \cos \left (c+\frac {3 d x}{2}\right )-5 \cos \left (3 c+\frac {5 d x}{2}\right )+\cos \left (3 c+\frac {7 d x}{2}\right )+131 \sin \left (\frac {d x}{2}\right )-72 d x \sin \left (c+\frac {d x}{2}\right )-27 \sin \left (2 c+\frac {3 d x}{2}\right )+5 \sin \left (2 c+\frac {5 d x}{2}\right )+\sin \left (4 c+\frac {7 d x}{2}\right )}{24 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
-1/24*(-72*d*x*Cos[(d*x)/2] - 31*Cos[c + (d*x)/2] - 27*Cos[c + (3*d*x)/2] - 5*Cos[3*c + (5*d*x)/2] + Cos[3*c + (7*d*x)/2] + 131*Sin[(d*x)/2] - 72*d* x*Sin[c + (d*x)/2] - 27*Sin[2*c + (3*d*x)/2] + 5*Sin[2*c + (5*d*x)/2] + Si n[4*c + (7*d*x)/2])/(a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[( c + d*x)/2]))
Time = 0.45 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3353, 3042, 3445, 2009}
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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^2(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^2}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3353 |
| \(\displaystyle \frac {\int \frac {\sin ^3(c+d x) (a-a \sin (c+d x))}{\sin (c+d x) a+a}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 (a-a \sin (c+d x))}{\sin (c+d x) a+a}dx}{a^2}\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \frac {\int \left (-\sin ^3(c+d x)+2 \sin ^2(c+d x)-2 \sin (c+d x)-\frac {2}{\sin (c+d x)+1}+2\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\cos ^3(c+d x)}{3 d}+\frac {3 \cos (c+d x)}{d}-\frac {\sin (c+d x) \cos (c+d x)}{d}+\frac {2 \cos (c+d x)}{d (\sin (c+d x)+1)}+3 x}{a^2}\) |
(3*x + (3*Cos[c + d*x])/d - Cos[c + d*x]^3/(3*d) - (Cos[c + d*x]*Sin[c + d *x])/d + (2*Cos[c + d*x])/(d*(1 + Sin[c + d*x])))/a^2
3.4.8.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/b^2 Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || !IGtQ[ n, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[si n[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{ a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ [m] && IntegerQ[n]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {3 x}{a^{2}}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {4}{d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(98\) |
| derivativedivides | \(\frac {\frac {4 \left (\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {4}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4}}{d \,a^{2}}\) | \(104\) |
| default | \(\frac {\frac {4 \left (\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {4}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4}}{d \,a^{2}}\) | \(104\) |
| parallelrisch | \(\frac {72 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+72 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-113 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+27 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+5 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-\sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-5 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+27 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{24 d \,a^{2} \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(145\) |
| norman | \(\frac {\frac {105 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {48 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {120 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {24 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {120 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {75 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {105 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {75 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {48 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {24 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {28}{3 a d}+\frac {9 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x}{a}+\frac {18 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {170 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {532 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {9 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {86 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {310 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {136 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {572 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {196 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {442 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {46 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(493\) |
3*x/a^2+11/8/d/a^2*exp(I*(d*x+c))+11/8/d/a^2*exp(-I*(d*x+c))+4/d/a^2/(exp( I*(d*x+c))+I)-1/12/d/a^2*cos(3*d*x+3*c)-1/2/d/a^2*sin(2*d*x+2*c)
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.43 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \, {\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) - 9 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{3} - 9 \, d x + 3 \, \cos \left (d x + c\right )^{2} - 6 \, \cos \left (d x + c\right ) + 6\right )} \sin \left (d x + c\right ) - 6}{3 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]
-1/3*(cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 9*d*x - 3*(3*d*x + 4)*cos(d*x + c) - 9*cos(d*x + c)^2 + (cos(d*x + c)^3 - 9*d*x + 3*cos(d*x + c)^2 - 6*cos (d*x + c) + 6)*sin(d*x + c) - 6)/(a^2*d*cos(d*x + c) + a^2*d*sin(d*x + c) + a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 2263 vs. \(2 (75) = 150\).
Time = 12.77 (sec) , antiderivative size = 2263, normalized size of antiderivative = 27.27 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((9*d*x*tan(c/2 + d*x/2)**7/(3*a**2*d*tan(c/2 + d*x/2)**7 + 3*a** 2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**5 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**3 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d*tan(c/2 + d*x/2) + 3*a**2*d) + 9*d*x*tan(c/2 + d*x/2)**6/(3*a** 2*d*tan(c/2 + d*x/2)**7 + 3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**5 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**3 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d*tan(c/2 + d*x/2) + 3*a**2*d) + 2 7*d*x*tan(c/2 + d*x/2)**5/(3*a**2*d*tan(c/2 + d*x/2)**7 + 3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**5 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**3 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d* tan(c/2 + d*x/2) + 3*a**2*d) + 27*d*x*tan(c/2 + d*x/2)**4/(3*a**2*d*tan(c/ 2 + d*x/2)**7 + 3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)** 5 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**3 + 9*a**2*d *tan(c/2 + d*x/2)**2 + 3*a**2*d*tan(c/2 + d*x/2) + 3*a**2*d) + 27*d*x*tan( c/2 + d*x/2)**3/(3*a**2*d*tan(c/2 + d*x/2)**7 + 3*a**2*d*tan(c/2 + d*x/2)* *6 + 9*a**2*d*tan(c/2 + d*x/2)**5 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2* d*tan(c/2 + d*x/2)**3 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d*tan(c/2 + d*x/2) + 3*a**2*d) + 27*d*x*tan(c/2 + d*x/2)**2/(3*a**2*d*tan(c/2 + d*x/2) **7 + 3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**5 + 9*a**2 *d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**3 + 9*a**2*d*tan(c/...
Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 312, normalized size of antiderivative = 3.76 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 14}{a^{2} + \frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
2/3*((5*sin(d*x + c)/(cos(d*x + c) + 1) + 33*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 18*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 24*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 9*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 9*sin(d*x + c)^6/ (cos(d*x + c) + 1)^6 + 14)/(a^2 + a^2*sin(d*x + c)/(cos(d*x + c) + 1) + 3* a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3*a^2*sin(d*x + c) ^5/(cos(d*x + c) + 1)^5 + a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^2*si n(d*x + c)^7/(cos(d*x + c) + 1)^7) + 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {9 \, {\left (d x + c\right )}}{a^{2}} + \frac {12}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]
1/3*(9*(d*x + c)/a^2 + 12/(a^2*(tan(1/2*d*x + 1/2*c) + 1)) + 2*(3*tan(1/2* d*x + 1/2*c)^5 + 6*tan(1/2*d*x + 1/2*c)^4 + 18*tan(1/2*d*x + 1/2*c)^2 - 3* tan(1/2*d*x + 1/2*c) + 8)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^2))/d
Time = 13.46 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3\,x}{a^2}+\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {28}{3}}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]