Integrand size = 27, antiderivative size = 47 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 x}{a^2}+\frac {\cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{d \left (a^2+a^2 \sin (c+d x)\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(117\) vs. \(2(47)=94\).
Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.49 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {12 d x \cos \left (\frac {d x}{2}\right )+2 \cos \left (c+\frac {d x}{2}\right )+3 \cos \left (c+\frac {3 d x}{2}\right )-28 \sin \left (\frac {d x}{2}\right )+12 d x \sin \left (c+\frac {d x}{2}\right )+3 \sin \left (2 c+\frac {3 d x}{2}\right )}{6 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(12*d*x*Cos[(d*x)/2] + 2*Cos[c + (d*x)/2] + 3*Cos[c + (3*d*x)/2] - 28*Sin[ (d*x)/2] + 12*d*x*Sin[c + (d*x)/2] + 3*Sin[2*c + (3*d*x)/2])/(6*a^2*d*(Cos [c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3336, 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.
\(\displaystyle \int \frac {\sin (c+d x) \cos ^2(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^2}{(a \sin (c+d x)+a)^2}dx\) |
\(\Big \downarrow \) 3336 |
\(\displaystyle \frac {2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {\int (a \sin (c+d x)-2 a)dx}{a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {-\frac {a \cos (c+d x)}{d}-2 a x}{a^3}\) |
3.4.10.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*( (c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos [e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Simp[1/(b^ 3*(2*m + 3)) Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d* (2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {4}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{2}}\) | \(50\) |
default | \(\frac {\frac {4}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{2}}\) | \(50\) |
parallelrisch | \(\frac {4 d x \cos \left (d x +c \right )-2 \cos \left (d x +c \right )-4 \sin \left (d x +c \right )+\cos \left (2 d x +2 c \right )+5}{2 d \,a^{2} \cos \left (d x +c \right )}\) | \(54\) |
risch | \(\frac {2 x}{a^{2}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{2}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{2}}+\frac {4}{d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\) | \(64\) |
norman | \(\frac {\frac {6}{a d}+\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 x}{a}+\frac {6 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {12 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {20 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {24 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {24 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {20 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {12 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {26 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {22 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {38 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {34 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {38 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(349\) |
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, d x + {\left (2 \, d x + 3\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + {\left (2 \, d x + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 2}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
(2*d*x + (2*d*x + 3)*cos(d*x + c) + cos(d*x + c)^2 + (2*d*x + cos(d*x + c) - 2)*sin(d*x + c) + 2)/(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. 479 vs. \(2 (41) = 82\).
Time = 3.80 (sec) , antiderivative size = 479, normalized size of antiderivative = 10.19 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\begin {cases} \frac {2 d x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {4 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {6}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos ^{2}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Piecewise((2*d*x*tan(c/2 + d*x/2)**3/(a**2*d*tan(c/2 + d*x/2)**3 + a**2*d* tan(c/2 + d*x/2)**2 + a**2*d*tan(c/2 + d*x/2) + a**2*d) + 2*d*x*tan(c/2 + d*x/2)**2/(a**2*d*tan(c/2 + d*x/2)**3 + a**2*d*tan(c/2 + d*x/2)**2 + a**2* d*tan(c/2 + d*x/2) + a**2*d) + 2*d*x*tan(c/2 + d*x/2)/(a**2*d*tan(c/2 + d* x/2)**3 + a**2*d*tan(c/2 + d*x/2)**2 + a**2*d*tan(c/2 + d*x/2) + a**2*d) + 2*d*x/(a**2*d*tan(c/2 + d*x/2)**3 + a**2*d*tan(c/2 + d*x/2)**2 + a**2*d*t an(c/2 + d*x/2) + a**2*d) + 4*tan(c/2 + d*x/2)**2/(a**2*d*tan(c/2 + d*x/2) **3 + a**2*d*tan(c/2 + d*x/2)**2 + a**2*d*tan(c/2 + d*x/2) + a**2*d) + 2*t an(c/2 + d*x/2)/(a**2*d*tan(c/2 + d*x/2)**3 + a**2*d*tan(c/2 + d*x/2)**2 + a**2*d*tan(c/2 + d*x/2) + a**2*d) + 6/(a**2*d*tan(c/2 + d*x/2)**3 + a**2* d*tan(c/2 + d*x/2)**2 + a**2*d*tan(c/2 + d*x/2) + a**2*d), Ne(d, 0)), (x*s in(c)*cos(c)**2/(a*sin(c) + a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.96 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 3}{a^{2} + \frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{d} \]
2*((sin(d*x + c)/(cos(d*x + c) + 1) + 2*sin(d*x + c)^2/(cos(d*x + c) + 1)^ 2 + 3)/(a^2 + a^2*sin(d*x + c)/(cos(d*x + c) + 1) + a^2*sin(d*x + c)^2/(co s(d*x + c) + 1)^2 + a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) + 2*arctan(si n(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.66 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {d x + c}{a^{2}} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} a^{2}}\right )}}{d} \]
2*((d*x + c)/a^2 + (2*tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c) + 3)/( (tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c) + 1)*a^2))/d
Time = 9.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2\,x}{a^2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]