Integrand size = 15, antiderivative size = 38 \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=-\frac {\sec ^2(c+b x) \sin (a-c)}{2 b}+\frac {\cos (a-c) \tan (c+b x)}{b} \]
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=-\frac {\sec (c) \sec ^2(c+b x) (\sin (a)-\cos (a-c) \sin (c+2 b x))}{2 b} \]
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5094, 3042, 3086, 15, 4254, 24}
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.
\(\displaystyle \int \cos (a+b x) \sec ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 5094 |
\(\displaystyle \cos (a-c) \int \sec ^2(c+b x)dx-\sin (a-c) \int \sec ^2(c+b x) \tan (c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )^2dx-\sin (a-c) \int \sec (c+b x)^2 \tan (c+b x)dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )^2dx-\frac {\sin (a-c) \int \sec (c+b x)d\sec (c+b x)}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )^2dx-\frac {\sin (a-c) \sec ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\cos (a-c) \int 1d(-\tan (c+b x))}{b}-\frac {\sin (a-c) \sec ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\cos (a-c) \tan (b x+c)}{b}-\frac {\sin (a-c) \sec ^2(b x+c)}{2 b}\) |
3.3.43.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[Cos[v_]*Sec[w_]^(n_.), x_Symbol] :> Simp[-Sin[v - w] Int[Tan[w]*Sec[w ]^(n - 1), x], x] + Simp[Cos[v - w] Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Time = 2.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {\sin \left (2 x b +a +c \right )}{b \left (\cos \left (2 x b +2 c \right )+1\right )}\) | \(26\) |
default | \(-\frac {1}{2 b \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \left (\tan \left (x b +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (x b +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}\) | \(56\) |
risch | \(\frac {i \left (2 \,{\mathrm e}^{i \left (2 x b +5 a +c \right )}+{\mathrm e}^{i \left (5 a -c \right )}+{\mathrm e}^{i \left (3 a +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}\) | \(61\) |
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) + \sin \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )^{2}} \]
Exception generated. \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (36) = 72\).
Time = 0.26 (sec) , antiderivative size = 382, normalized size of antiderivative = 10.05 \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=-\frac {{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) + {\left (\sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \, \cos \left (a + c\right ) \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) - {\left (\cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (a + c\right ) - 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} + 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) + b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) + b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \]
-((2*sin(2*b*x + 2*a + 2*c) + sin(2*a) + sin(2*c))*cos(4*b*x + a + 5*c) + 2*(2*sin(2*b*x + 2*a + 2*c) + sin(2*a) + sin(2*c))*cos(2*b*x + a + 3*c) + (sin(2*a) + sin(2*c))*cos(a + c) - (2*cos(2*b*x + 2*a + 2*c) + cos(2*a) + cos(2*c))*sin(4*b*x + a + 5*c) + 2*cos(a + c)*sin(2*b*x + 2*a + 2*c) - 2*( 2*cos(2*b*x + 2*a + 2*c) + cos(2*a) + cos(2*c))*sin(2*b*x + a + 3*c) - (co s(2*a) + cos(2*c))*sin(a + c) - 2*cos(2*b*x + 2*a + 2*c)*sin(a + c))/(b*co s(4*b*x + a + 5*c)^2 + 4*b*cos(2*b*x + a + 3*c)^2 + 4*b*cos(2*b*x + a + 3* c)*cos(a + c) + b*cos(a + c)^2 + b*sin(4*b*x + a + 5*c)^2 + 4*b*sin(2*b*x + a + 3*c)^2 + 4*b*sin(2*b*x + a + 3*c)*sin(a + c) + b*sin(a + c)^2 + 2*(2 *b*cos(2*b*x + a + 3*c) + b*cos(a + c))*cos(4*b*x + a + 5*c) + 2*(2*b*sin( 2*b*x + a + 3*c) + b*sin(a + c))*sin(4*b*x + a + 5*c))
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (36) = 72\).
Time = 0.34 (sec) , antiderivative size = 315, normalized size of antiderivative = 8.29 \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=-\frac {\tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{6} + \tan \left (\frac {1}{2} \, a\right )^{6} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{2} + 1}{4 \, {\left (2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )}^{2} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} b} \]
-1/4*(tan(1/2*a)^6*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 3*tan(1/2* a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan(1/2*a)^4*tan(1/2*c )^4 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)^6 + 9*tan(1/2*a)^4*tan(1/2* c)^2 + 9*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*c)^6 + 3*tan(1/2*a)^4 + 9*tan (1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*c)^4 + 3*tan(1/2*a)^2 + 3*tan(1/2*c)^2 + 1)/((2*tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(b*x + a)*tan(1/2*a)* tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan(b*x + a)*tan(1/2*a) - tan (1/2*a)^2 - 2*tan(b*x + a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) - tan(1/2* c)^2 + 1)^2*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a ) - tan(1/2*c))*b)
Timed out. \[ \int \cos (a+b x) \sec ^3(c+b x) \, dx=\text {Hanged} \]