Integrand size = 31, antiderivative size = 182 \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=-\frac {11 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}}\right )}{16 b}+\frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {11 c^2 \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}} \]
-11/16*c^(3/2)*arctanh(c^(1/2)*tan(2*b*x+2*a)/(-c+c*sec(2*b*x+2*a))^(1/2)) /b+11/16*c^2*sin(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)-11/24*c^2*cos(2* b*x+2*a)*sin(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)+1/6*c^2*cos(2*b*x+2* a)^2*sin(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)
Time = 1.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.64 \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {c \left (38 \cot (a+b x)-33 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \cos (a+b x)}{\sqrt {\cos (2 (a+b x))}}\right ) \sqrt {\cos (2 (a+b x))} \csc (a+b x)-42 \sin (2 (a+b x))+14 \sin (4 (a+b x))-4 \sin (6 (a+b x))\right ) \sqrt {c \tan (a+b x) \tan (2 (a+b x))}}{96 b} \]
(c*(38*Cot[a + b*x] - 33*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2 *(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x] - 42*Sin[2*(a + b*x)] + 14*Sin[4*(a + b*x)] - 4*Sin[6*(a + b*x)])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b *x)]])/(96*b)
Time = 0.89 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4897, 3042, 4300, 27, 2011, 3042, 4292, 3042, 4292, 3042, 4261, 220}
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.
| \(\displaystyle \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (2 a+2 b x)^3 (c \tan (a+b x) \tan (2 a+2 b x))^{3/2}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \cos ^3(2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c\right )^{3/2}}{\csc \left (2 a+2 b x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4300 |
| \(\displaystyle \frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {1}{3} c \int \frac {11 \cos ^2(2 a+2 b x) (c-c \sec (2 a+2 b x))}{2 \sqrt {c \sec (2 a+2 b x)-c}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {11}{6} c \int \frac {\cos ^2(2 a+2 b x) (c-c \sec (2 a+2 b x))}{\sqrt {c \sec (2 a+2 b x)-c}}dx\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle \frac {11}{6} c \int \cos ^2(2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}dx+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{6} c \int \frac {\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}{\csc \left (2 a+2 b x+\frac {\pi }{2}\right )^2}dx+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \int \cos (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}dx-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \int \frac {\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}{\csc \left (2 a+2 b x+\frac {\pi }{2}\right )}dx-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \left (-\frac {1}{2} \int \sqrt {c \sec (2 a+2 b x)-c}dx-\frac {c \sin (2 a+2 b x)}{2 b \sqrt {c \sec (2 a+2 b x)-c}}\right )-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \left (-\frac {1}{2} \int \sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}dx-\frac {c \sin (2 a+2 b x)}{2 b \sqrt {c \sec (2 a+2 b x)-c}}\right )-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \left (\frac {c \int \frac {1}{\frac {c^2 \tan ^2(2 a+2 b x)}{c \sec (2 a+2 b x)-c}-c}d\left (-\frac {c \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{2 b}-\frac {c \sin (2 a+2 b x)}{2 b \sqrt {c \sec (2 a+2 b x)-c}}\right )-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {11}{6} c \left (-\frac {3}{4} \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{2 b}-\frac {c \sin (2 a+2 b x)}{2 b \sqrt {c \sec (2 a+2 b x)-c}}\right )-\frac {c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt {c \sec (2 a+2 b x)-c}}\right )+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}\) |
(c^2*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x ]]) + (11*c*(-1/4*(c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec [2*a + 2*b*x]]) - (3*((Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b) - (c*Sin[2*a + 2*b*x])/(2*b*Sqrt[-c + c*Sec[ 2*a + 2*b*x]])))/4))/6
3.7.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a*((2*n + 1)/(2*b*d*n)) Int[Sqrt[a + b*Csc [e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)* ((d*Csc[e + f*x])^n/(f*n)), x] - Simp[a/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*(b*(m - 2*n - 2) - a*(m + 2*n - 1)*Csc[e + f *x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && (LtQ[n, -1] || (EqQ[m, 3/2] && EqQ[n, -2^(-1)])) && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(162)=324\).
Time = 6.48 (sec) , antiderivative size = 958, normalized size of antiderivative = 5.26
1/2*2^(1/2)/b*c*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*(cot(b*x+a)*((2* cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*arctanh(cos(b*x+a)/(1+cos(b*x+a))/ ((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*2^(1/2))*2^(1/2)+csc(b*x+a)*(( 2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*arctanh(cos(b*x+a)/(1+cos(b*x+a) )/((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*2^(1/2))*2^(1/2)+2*cot(b*x+a ))*4^(1/2)/(2+2^(1/2))/(-2+2^(1/2))+3*2^(1/2)/b*csc(b*x+a)*(2^(1/2)*arctan h(cos(b*x+a)/(1+cos(b*x+a))/((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*2^ (1/2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*cos(b*x+a)+2^(1/2)*arct anh(cos(b*x+a)/(1+cos(b*x+a))/((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)* 2^(1/2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)-4*cos(b*x+a)^3-2*cos( b*x+a))*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*c*4^(1/2)/(-2+2^(1/2))^3 /(2+2^(1/2))^3-3*2^(1/2)/b*csc(b*x+a)*(-16*cos(b*x+a)^5+9*2^(1/2)*arctanh( cos(b*x+a)/(1+cos(b*x+a))/((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*2^(1 /2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*cos(b*x+a)+9*2^(1/2)*arct anh(cos(b*x+a)/(1+cos(b*x+a))/((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)* 2^(1/2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)+12*cos(b*x+a)^3-18*co s(b*x+a))*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*c*4^(1/2)/(-2+2^(1/2)) ^5/(2+2^(1/2))^5+2/3*2^(1/2)/b*csc(b*x+a)*(-128*cos(b*x+a)^7+80*cos(b*x+a) ^5+75*2^(1/2)*arctanh(cos(b*x+a)/(1+cos(b*x+a))/((2*cos(b*x+a)^2-1)/(1+cos (b*x+a))^2)^(1/2)*2^(1/2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*...
Time = 0.28 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.76 \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\left [\frac {33 \, {\left (c \tan \left (b x + a\right )^{7} + 3 \, c \tan \left (b x + a\right )^{5} + 3 \, c \tan \left (b x + a\right )^{3} + c \tan \left (b x + a\right )\right )} \sqrt {c} \log \left (-\frac {c \tan \left (b x + a\right )^{5} - 14 \, c \tan \left (b x + a\right )^{3} - 4 \, \sqrt {2} {\left (\tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} + 3\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} \sqrt {c} + 17 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{5} + 2 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}\right ) - 4 \, \sqrt {2} {\left (63 \, c \tan \left (b x + a\right )^{6} - 13 \, c \tan \left (b x + a\right )^{4} - 31 \, c \tan \left (b x + a\right )^{2} - 19 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{192 \, {\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}, \frac {33 \, {\left (c \tan \left (b x + a\right )^{7} + 3 \, c \tan \left (b x + a\right )^{5} + 3 \, c \tan \left (b x + a\right )^{3} + c \tan \left (b x + a\right )\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-c}}{c \tan \left (b x + a\right )^{3} - 3 \, c \tan \left (b x + a\right )}\right ) - 2 \, \sqrt {2} {\left (63 \, c \tan \left (b x + a\right )^{6} - 13 \, c \tan \left (b x + a\right )^{4} - 31 \, c \tan \left (b x + a\right )^{2} - 19 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{96 \, {\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}\right ] \]
[1/192*(33*(c*tan(b*x + a)^7 + 3*c*tan(b*x + a)^5 + 3*c*tan(b*x + a)^3 + c *tan(b*x + a))*sqrt(c)*log(-(c*tan(b*x + a)^5 - 14*c*tan(b*x + a)^3 - 4*sq rt(2)*(tan(b*x + a)^4 - 4*tan(b*x + a)^2 + 3)*sqrt(-c*tan(b*x + a)^2/(tan( b*x + a)^2 - 1))*sqrt(c) + 17*c*tan(b*x + a))/(tan(b*x + a)^5 + 2*tan(b*x + a)^3 + tan(b*x + a))) - 4*sqrt(2)*(63*c*tan(b*x + a)^6 - 13*c*tan(b*x + a)^4 - 31*c*tan(b*x + a)^2 - 19*c)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)))/(b*tan(b*x + a)^7 + 3*b*tan(b*x + a)^5 + 3*b*tan(b*x + a)^3 + b*tan (b*x + a)), 1/96*(33*(c*tan(b*x + a)^7 + 3*c*tan(b*x + a)^5 + 3*c*tan(b*x + a)^3 + c*tan(b*x + a))*sqrt(-c)*arctan(2*sqrt(2)*sqrt(-c*tan(b*x + a)^2/ (tan(b*x + a)^2 - 1))*(tan(b*x + a)^2 - 1)*sqrt(-c)/(c*tan(b*x + a)^3 - 3* c*tan(b*x + a))) - 2*sqrt(2)*(63*c*tan(b*x + a)^6 - 13*c*tan(b*x + a)^4 - 31*c*tan(b*x + a)^2 - 19*c)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)))/ (b*tan(b*x + a)^7 + 3*b*tan(b*x + a)^5 + 3*b*tan(b*x + a)^3 + b*tan(b*x + a))]
Timed out. \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\int {\cos \left (2\,a+2\,b\,x\right )}^3\,{\left (c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )\right )}^{3/2} \,d x \]