Integrand size = 21, antiderivative size = 58 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=a b x+\frac {\left (a^2+b^2\right ) \sin (c+d x)}{d}+\frac {a b \cos (c+d x) \sin (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x)}{3 d} \]
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {3 \left (3 a^2+4 b^2\right ) \sin (c+d x)+a (12 b (c+d x)+6 b \sin (2 (c+d x))+a \sin (3 (c+d x)))}{12 d} \]
(3*(3*a^2 + 4*b^2)*Sin[c + d*x] + a*(12*b*(c + d*x) + 6*b*Sin[2*(c + d*x)] + a*Sin[3*(c + d*x)]))/(12*d)
Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4275, 3042, 3115, 24, 4532, 3042, 3492, 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.
\(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 4275 |
\(\displaystyle \int \cos ^3(c+d x) \left (a^2+b^2 \sec ^2(c+d x)\right )dx+2 a b \int \cos ^2(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a^2+b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+2 a b \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \int \frac {a^2+b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+2 a b \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \int \frac {a^2+b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+2 a b \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\) |
\(\Big \downarrow \) 4532 |
\(\displaystyle \int \cos (c+d x) \left (b^2+a^2 \cos ^2(c+d x)\right )dx+2 a b \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (b^2+a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+2 a b \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\) |
\(\Big \downarrow \) 3492 |
\(\displaystyle 2 a b \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (-\sin ^2(c+d x) a^2+a^2+b^2\right )d(-\sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a b \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\frac {1}{3} a^2 \sin ^3(c+d x)-\left (a^2+b^2\right ) \sin (c+d x)}{d}\) |
2*a*b*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (-((a^2 + b^2)*Sin[c + d *x]) + (a^2*Sin[c + d*x]^3)/3)/d
3.5.63.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[2*a*(b/d) Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
Time = 0.64 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(\frac {12 a b x d +9 a^{2} \sin \left (d x +c \right )+12 b^{2} \sin \left (d x +c \right )+a^{2} \sin \left (3 d x +3 c \right )+6 a b \sin \left (2 d x +2 c \right )}{12 d}\) | \(61\) |
derivativedivides | \(\frac {\frac {a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} \sin \left (d x +c \right )}{d}\) | \(63\) |
default | \(\frac {\frac {a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} \sin \left (d x +c \right )}{d}\) | \(63\) |
risch | \(a b x +\frac {3 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) b^{2}}{d}+\frac {a^{2} \sin \left (3 d x +3 c \right )}{12 d}+\frac {a b \sin \left (2 d x +2 c \right )}{2 d}\) | \(66\) |
norman | \(\frac {a b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-a b x -\frac {2 \left (a^{2}-3 a b -3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {2 \left (a^{2}-a b +b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {2 \left (a^{2}+a b +b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (a^{2}+3 a b -3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-2 a b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(194\) |
1/12*(12*a*b*x*d+9*a^2*sin(d*x+c)+12*b^2*sin(d*x+c)+a^2*sin(3*d*x+3*c)+6*a *b*sin(2*d*x+2*c))/d
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {3 \, a b d x + {\left (a^{2} \cos \left (d x + c\right )^{2} + 3 \, a b \cos \left (d x + c\right ) + 2 \, a^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cos ^{3}{\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b - 6 \, b^{2} \sin \left (d x + c\right )}{6 \, d} \]
-1/6*(2*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^2 - 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*a*b - 6*b^2*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (56) = 112\).
Time = 0.30 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.64 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {3 \, {\left (d x + c\right )} a b + \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
1/3*(3*(d*x + c)*a*b + 2*(3*a^2*tan(1/2*d*x + 1/2*c)^5 - 3*a*b*tan(1/2*d*x + 1/2*c)^5 + 3*b^2*tan(1/2*d*x + 1/2*c)^5 + 2*a^2*tan(1/2*d*x + 1/2*c)^3 + 6*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*a^2*tan(1/2*d*x + 1/2*c) + 3*a*b*tan(1/ 2*d*x + 1/2*c) + 3*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^ 3)/d
Time = 13.44 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.24 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {2\,a^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {b^2\,\sin \left (c+d\,x\right )}{d}+a\,b\,x+\frac {a^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \]