Integrand size = 27, antiderivative size = 45 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{2 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d} \]
Leaf count is larger than twice the leaf count of optimal. \(161\) vs. \(2(45)=90\).
Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.58 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 (c-2 d x) \cos \left (\frac {c}{2}\right )-4 \cos \left (\frac {c}{2}+d x\right )-4 \cos \left (\frac {3 c}{2}+d x\right )+\cos \left (\frac {3 c}{2}+2 d x\right )-\cos \left (\frac {5 c}{2}+2 d x\right )-4 \sin \left (\frac {c}{2}\right )+2 c \sin \left (\frac {c}{2}\right )-4 d x \sin \left (\frac {c}{2}\right )+4 \sin \left (\frac {c}{2}+d x\right )-4 \sin \left (\frac {3 c}{2}+d x\right )+\sin \left (\frac {3 c}{2}+2 d x\right )+\sin \left (\frac {5 c}{2}+2 d x\right )}{8 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
(2*(c - 2*d*x)*Cos[c/2] - 4*Cos[c/2 + d*x] - 4*Cos[(3*c)/2 + d*x] + Cos[(3 *c)/2 + 2*d*x] - Cos[(5*c)/2 + 2*d*x] - 4*Sin[c/2] + 2*c*Sin[c/2] - 4*d*x* Sin[c/2] + 4*Sin[c/2 + d*x] - 4*Sin[(3*c)/2 + d*x] + Sin[(3*c)/2 + 2*d*x] + Sin[(5*c)/2 + 2*d*x])/(8*a*d*(Cos[c/2] + Sin[c/2]))
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 3318, 3042, 3115, 24, 3118}
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.
\(\displaystyle \int \frac {\sin (c+d x) \cos ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^2}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int \sin (c+d x)dx}{a}-\frac {\int \sin ^2(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sin (c+d x)dx}{a}-\frac {\int \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\int \sin (c+d x)dx}{a}-\frac {\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\int \sin (c+d x)dx}{a}-\frac {\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {\cos (c+d x)}{a d}-\frac {\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}}{a}\) |
3.3.100.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71
method | result | size |
parallelrisch | \(\frac {-2 d x -4 \cos \left (d x +c \right )+\sin \left (2 d x +2 c \right )+4}{4 d a}\) | \(32\) |
risch | \(-\frac {x}{2 a}-\frac {\cos \left (d x +c \right )}{a d}+\frac {\sin \left (2 d x +2 c \right )}{4 d a}\) | \(39\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {1}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(77\) |
default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {1}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(77\) |
norman | \(\frac {-\frac {1}{a d}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {x}{2 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {3 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(217\) |
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )}{2 \, a d} \]
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (32) = 64\).
Time = 1.89 (sec) , antiderivative size = 366, normalized size of antiderivative = 8.13 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} - \frac {d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {2 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {d x}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {2 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {4 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {4}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos ^{2}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
Piecewise((-d*x*tan(c/2 + d*x/2)**4/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan (c/2 + d*x/2)**2 + 2*a*d) - 2*d*x*tan(c/2 + d*x/2)**2/(2*a*d*tan(c/2 + d*x /2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - d*x/(2*a*d*tan(c/2 + d*x/2)* *4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - 2*tan(c/2 + d*x/2)**3/(2*a*d*tan (c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - 4*tan(c/2 + d*x/2) **2/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) + 2*ta n(c/2 + d*x/2)/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2* a*d) - 4/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d), Ne(d, 0)), (x*sin(c)*cos(c)**2/(a*sin(c) + a), True))
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (41) = 82\).
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.96 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 2}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \]
((sin(d*x + c)/(cos(d*x + c) + 1) - 2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 2)/(a + 2*a*sin(d*x + c)^2/(cos(d* x + c) + 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {d x + c}{a} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
-1/2*((d*x + c)/a + 2*(tan(1/2*d*x + 1/2*c)^3 + 2*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a))/d
Time = 9.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{2\,a}-\frac {\cos \left (c+d\,x\right )-\frac {\sin \left (2\,c+2\,d\,x\right )}{4}}{a\,d} \]