Integrand size = 31, antiderivative size = 110 \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b} \]
1/5*(-c+c*sec(2*b*x+2*a))^(3/2)*tan(2*b*x+2*a)/b+4/5*c^2*tan(2*b*x+2*a)/b/ (-c+c*sec(2*b*x+2*a))^(1/2)-1/5*c*(-c+c*sec(2*b*x+2*a))^(1/2)*tan(2*b*x+2* a)/b
Time = 0.63 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {\cot (a+b x) (-2+4 \cot (a+b x) \cot (2 (a+b x))+\sec (2 (a+b x))) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}{5 b} \]
(Cot[a + b*x]*(-2 + 4*Cot[a + b*x]*Cot[2*(a + b*x)] + Sec[2*(a + b*x)])*(c *Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/(5*b)
Time = 0.62 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3042, 4897, 3042, 4285, 3042, 4280, 3042, 4279}
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.
\(\displaystyle \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (2 a+2 b x)^2 (c \tan (a+b x) \tan (2 a+2 b x))^{3/2}dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int \sec ^2(2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (2 a+2 b x+\frac {\pi }{2}\right )^2 \left (c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c\right )^{3/2}dx\) |
\(\Big \downarrow \) 4285 |
\(\displaystyle \frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}-\frac {3}{5} \int \sec (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}-\frac {3}{5} \int \csc \left (2 a+2 b x+\frac {\pi }{2}\right ) \left (c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c\right )^{3/2}dx\) |
\(\Big \downarrow \) 4280 |
\(\displaystyle \frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}-\frac {3}{5} \left (\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b}-\frac {4}{3} c \int \sec (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}-\frac {3}{5} \left (\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b}-\frac {4}{3} c \int \csc \left (2 a+2 b x+\frac {\pi }{2}\right ) \sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}dx\right )\) |
\(\Big \downarrow \) 4279 |
\(\displaystyle \frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}-\frac {3}{5} \left (\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b}-\frac {4 c^2 \tan (2 a+2 b x)}{3 b \sqrt {c \sec (2 a+2 b x)-c}}\right )\) |
((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b) - (3*((-4*c^2*Tan [2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sqrt[-c + c*Sec[2* a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b)))/5
3.7.13.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ Symbol] :> Simp[(-b)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[a*((2*m - 1)/m) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && Intege rQ[2*m]
Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[a*(m/(b*(m + 1))) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 5.67 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\sqrt {2}\, \cot \left (x b +a \right ) \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \left (12 \cos \left (x b +a \right )^{4}-15 \cos \left (x b +a \right )^{2}+5\right ) c \sqrt {4}}{5 b \left (2 \cos \left (x b +a \right )^{2}-1\right )^{2}}\) | \(81\) |
1/5*2^(1/2)/b*cot(b*x+a)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*(12*cos (b*x+a)^4-15*cos(b*x+a)^2+5)*c/(2*cos(b*x+a)^2-1)^2*4^(1/2)
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.80 \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {2 \, \sqrt {2} {\left (5 \, c \tan \left (b x + a\right )^{4} - 5 \, c \tan \left (b x + a\right )^{2} + 2 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{5 \, {\left (b \tan \left (b x + a\right )^{5} - 2 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}} \]
2/5*sqrt(2)*(5*c*tan(b*x + a)^4 - 5*c*tan(b*x + a)^2 + 2*c)*sqrt(-c*tan(b* x + a)^2/(tan(b*x + a)^2 - 1))/(b*tan(b*x + a)^5 - 2*b*tan(b*x + a)^3 + b* tan(b*x + a))
Timed out. \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out} \]
\[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\int { \left (c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sec \left (2 \, b x + 2 \, a\right )^{2} \,d x } \]
-2/5*(10*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1 )^(1/4)*((b*c*cos(4*b*x + 4*a)^2 + b*c*sin(4*b*x + 4*a)^2 + 2*b*c*cos(4*b* x + 4*a) + b*c)*integrate(-(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*co s(4*b*x + 4*a) + 1)^(1/4)*(((cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8 *b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(12*b*x + 12*a)*sin (4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)* cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4 *a)*sin(12*b*x + 12*a) + 2*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 2*cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*sin(3/2*ar ctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*cos(5/2*arctan2(sin(4*b*x + 4*a), cos(4*b*x + 4*a))) + ((cos(4*b*x + 4*a)*sin(12*b*x + 12*a) + 2*co s(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 2* cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos( 4*b*x + 4*a) - 1)) - (cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(12*b*x + 12*a)*sin(4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(3/2 *arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sin(5/2*arctan2(sin(4* b*x + 4*a), cos(4*b*x + 4*a))))/((cos(4*b*x + 4*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(1 2*b*x + 12*a)^2 + 4*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*...
Timed out. \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out} \]
Time = 45.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35 \[ \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {2\,c\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{5\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}^2} \]