Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {11 x}{2 a^3}+\frac {3 \sin (c+d x)}{a^3 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac {19 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))} \]
-11/2*x/a^3+3*sin(d*x+c)/a^3/d-1/2*cos(d*x+c)*sin(d*x+c)/a^3/d-2/3*sin(d*x +c)/a^3/d/(1+cos(d*x+c))^2+19/3*sin(d*x+c)/a^3/d/(1+cos(d*x+c))
Time = 0.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.82 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (1980 d x \cos \left (\frac {d x}{2}\right )+1980 d x \cos \left (c+\frac {d x}{2}\right )+660 d x \cos \left (c+\frac {3 d x}{2}\right )+660 d x \cos \left (2 c+\frac {3 d x}{2}\right )-3216 \sin \left (\frac {d x}{2}\right )+1326 \sin \left (c+\frac {d x}{2}\right )-2012 \sin \left (c+\frac {3 d x}{2}\right )-498 \sin \left (2 c+\frac {3 d x}{2}\right )-135 \sin \left (2 c+\frac {5 d x}{2}\right )-135 \sin \left (3 c+\frac {5 d x}{2}\right )+15 \sin \left (3 c+\frac {7 d x}{2}\right )+15 \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{960 a^3 d} \]
-1/960*(Sec[c/2]*Sec[(c + d*x)/2]^3*(1980*d*x*Cos[(d*x)/2] + 1980*d*x*Cos[ c + (d*x)/2] + 660*d*x*Cos[c + (3*d*x)/2] + 660*d*x*Cos[2*c + (3*d*x)/2] - 3216*Sin[(d*x)/2] + 1326*Sin[c + (d*x)/2] - 2012*Sin[c + (3*d*x)/2] - 498 *Sin[2*c + (3*d*x)/2] - 135*Sin[2*c + (5*d*x)/2] - 135*Sin[3*c + (5*d*x)/2 ] + 15*Sin[3*c + (7*d*x)/2] + 15*Sin[4*c + (7*d*x)/2]))/(a^3*d)
Time = 0.59 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4360, 25, 25, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^2}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^2(c+d x) \cos ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^2(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3353 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a-a \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{a^2}\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \frac {\int \left (-\frac {\cos ^2(c+d x)}{a}+\frac {3 \cos (c+d x)}{a}-\frac {5}{a}+\frac {7}{a (\cos (c+d x)+1)}-\frac {2}{a (\cos (c+d x)+1)^2}\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \sin (c+d x)}{a d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a d}+\frac {19 \sin (c+d x)}{3 a d (\cos (c+d x)+1)}-\frac {2 \sin (c+d x)}{3 a d (\cos (c+d x)+1)^2}-\frac {11 x}{2 a}}{a^2}\) |
((-11*x)/(2*a) + (3*Sin[c + d*x])/(a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a *d) - (2*Sin[c + d*x])/(3*a*d*(1 + Cos[c + d*x])^2) + (19*Sin[c + d*x])/(3 *a*d*(1 + Cos[c + d*x])))/a^2
3.2.3.3.1 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
| method | result | size |
| parallelrisch | \(\frac {\frac {275 \left (\cos \left (d x +c \right )+\frac {24 \cos \left (2 d x +2 c \right )}{275}-\frac {3 \cos \left (3 d x +3 c \right )}{275}+\frac {232}{275}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{48}-\frac {11 d x}{2}}{a^{3} d}\) | \(66\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(87\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(87\) |
| risch | \(-\frac {11 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {2 i \left (21 \,{\mathrm e}^{2 i \left (d x +c \right )}+36 \,{\mathrm e}^{i \left (d x +c \right )}+19\right )}{3 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(126\) |
| norman | \(\frac {-\frac {11 x}{2 a}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}-\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{2}}\) | \(135\) |
11/48*(25*(cos(d*x+c)+24/275*cos(2*d*x+2*c)-3/275*cos(3*d*x+3*c)+232/275)* tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^2-24*d*x)/a^3/d
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {33 \, d x \cos \left (d x + c\right )^{2} + 66 \, d x \cos \left (d x + c\right ) + 33 \, d x + {\left (3 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} - 71 \, \cos \left (d x + c\right ) - 52\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
-1/6*(33*d*x*cos(d*x + c)^2 + 66*d*x*cos(d*x + c) + 33*d*x + (3*cos(d*x + c)^3 - 12*cos(d*x + c)^2 - 71*cos(d*x + c) - 52)*sin(d*x + c))/(a^3*d*cos( d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)
\[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {18 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}} - \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
1/3*(3*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (18*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^3 - 33*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.35 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} - \frac {6 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {2 \, {\left (a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{9}}}{6 \, d} \]
-1/6*(33*(d*x + c)/a^3 - 6*(7*tan(1/2*d*x + 1/2*c)^3 + 5*tan(1/2*d*x + 1/2 *c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3) + 2*(a^6*tan(1/2*d*x + 1/2*c)^3 - 18*a^6*tan(1/2*d*x + 1/2*c))/a^9)/d
Time = 13.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-38\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-42\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+33\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{6\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]