Integrand size = 31, antiderivative size = 128 \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=-\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {-c+c \sec (2 a+2 b x)}}\right )}{4 \sqrt {2} b c^{3/2}}-\frac {\tan (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}+\frac {\tan (2 a+2 b x)}{b c \sqrt {-c+c \sec (2 a+2 b x)}} \]
-7/8*arctanh(1/2*c^(1/2)*tan(2*b*x+2*a)*2^(1/2)/(-c+c*sec(2*b*x+2*a))^(1/2 ))/b/c^(3/2)*2^(1/2)-1/4*tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(3/2)+tan( 2*b*x+2*a)/b/c/(-c+c*sec(2*b*x+2*a))^(1/2)
Time = 1.66 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\frac {\left (-5+4 \sec (2 (a+b x))+7 \arctan \left (\sqrt {-1+\tan ^2(a+b x)}\right ) \sec (2 (a+b x)) \sin ^2(a+b x) \sqrt {-1+\tan ^2(a+b x)}\right ) \tan (2 (a+b x))}{4 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \]
((-5 + 4*Sec[2*(a + b*x)] + 7*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sec[2*(a + b*x)]*Sin[a + b*x]^2*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*(a + b*x)])/(4*b*(c *Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))
Time = 0.70 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4897, 3042, 4286, 27, 3042, 4489, 3042, 4282, 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.
| \(\displaystyle \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (2 a+2 b x)^3}{(c \tan (a+b x) \tan (2 a+2 b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {\sec ^3(2 a+2 b x)}{(c \sec (2 a+2 b x)-c)^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (2 a+2 b x+\frac {\pi }{2}\right )^3}{\left (c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4286 |
| \(\displaystyle \frac {\int \frac {\sec (2 a+2 b x) (4 \sec (2 a+2 b x) c+3 c)}{2 \sqrt {c \sec (2 a+2 b x)-c}}dx}{2 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec (2 a+2 b x) (4 \sec (2 a+2 b x) c+3 c)}{\sqrt {c \sec (2 a+2 b x)-c}}dx}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (2 a+2 b x+\frac {\pi }{2}\right ) \left (4 \csc \left (2 a+2 b x+\frac {\pi }{2}\right ) c+3 c\right )}{\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}dx}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 4489 |
| \(\displaystyle \frac {7 c \int \frac {\sec (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}dx+\frac {4 c \tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 c \int \frac {\csc \left (2 a+2 b x+\frac {\pi }{2}\right )}{\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}dx+\frac {4 c \tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {4 c \tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {7 c \int \frac {1}{\frac {c^2 \tan ^2(2 a+2 b x)}{c \sec (2 a+2 b x)-c}-2 c}d\left (-\frac {c \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{b}}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\frac {4 c \tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {c \sec (2 a+2 b x)-c}}\right )}{\sqrt {2} b}}{4 c^2}-\frac {\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}\) |
-1/4*Tan[2*a + 2*b*x]/(b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) + ((-7*Sqrt[c]*A rcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])] )/(Sqrt[2]*b) + (4*c*Tan[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]]))/ (4*c^2)
3.7.27.3.1 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[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[b*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1) *(a*m - b*(2*m + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[(a*B*m + A*b*(m + 1))/(b*(m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B , e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b *(m + 1), 0] && !LtQ[m, -2^(-1)]
Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(111)=222\).
Time = 3.91 (sec) , antiderivative size = 495, normalized size of antiderivative = 3.87
| method | result | size |
| default | \(\frac {\sqrt {2}\, \csc \left (x b +a \right ) \left (1-\cos \left (x b +a \right )\right ) \left (\csc \left (x b +a \right )^{6} \left (1-\cos \left (x b +a \right )\right )^{6}+14 \csc \left (x b +a \right )^{2} \ln \left (\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}-3+\sqrt {\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1}\right ) \left (1-\cos \left (x b +a \right )\right )^{2} \sqrt {\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1}+14 \csc \left (x b +a \right )^{2} \operatorname {arctanh}\left (\frac {3 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}-1}{\sqrt {\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1}}\right ) \left (1-\cos \left (x b +a \right )\right )^{2} \sqrt {\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1}-39 \csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}+39 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}-1\right ) \sqrt {4}}{64 b \left (\frac {\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2} c}{\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1}\right )^{\frac {3}{2}} \left (\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-6 \csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1\right )^{2}}\) | \(495\) |
1/64*2^(1/2)/b*csc(b*x+a)/(csc(b*x+a)^2*(1-cos(b*x+a))^2*c/(csc(b*x+a)^4*( 1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-cos(b*x+a))^2+1))^(3/2)*(1-cos(b*x+a))*( csc(b*x+a)^6*(1-cos(b*x+a))^6+14*csc(b*x+a)^2*ln(csc(b*x+a)^2*(1-cos(b*x+a ))^2-3+(csc(b*x+a)^4*(1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-cos(b*x+a))^2+1)^( 1/2))*(1-cos(b*x+a))^2*(csc(b*x+a)^4*(1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-co s(b*x+a))^2+1)^(1/2)+14*csc(b*x+a)^2*arctanh((3*csc(b*x+a)^2*(1-cos(b*x+a) )^2-1)/(csc(b*x+a)^4*(1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-cos(b*x+a))^2+1)^( 1/2))*(1-cos(b*x+a))^2*(csc(b*x+a)^4*(1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-co s(b*x+a))^2+1)^(1/2)-39*csc(b*x+a)^4*(1-cos(b*x+a))^4+39*csc(b*x+a)^2*(1-c os(b*x+a))^2-1)/(csc(b*x+a)^4*(1-cos(b*x+a))^4-6*csc(b*x+a)^2*(1-cos(b*x+a ))^2+1)^2*4^(1/2)
Time = 0.28 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.16 \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\left [\frac {7 \, \sqrt {2} \sqrt {c} \log \left (\frac {c \tan \left (b x + a\right )^{3} - 2 \, \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {c} - 2 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3}}\right ) \tan \left (b x + a\right )^{3} + 2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (9 \, \tan \left (b x + a\right )^{2} - 1\right )}}{16 \, b c^{2} \tan \left (b x + a\right )^{3}}, -\frac {7 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-c}}{c \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right )^{3} - \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (9 \, \tan \left (b x + a\right )^{2} - 1\right )}}{8 \, b c^{2} \tan \left (b x + a\right )^{3}}\right ] \]
[1/16*(7*sqrt(2)*sqrt(c)*log((c*tan(b*x + a)^3 - 2*sqrt(-c*tan(b*x + a)^2/ (tan(b*x + a)^2 - 1))*(tan(b*x + a)^2 - 1)*sqrt(c) - 2*c*tan(b*x + a))/tan (b*x + a)^3)*tan(b*x + a)^3 + 2*sqrt(2)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1))*(9*tan(b*x + a)^2 - 1))/(b*c^2*tan(b*x + a)^3), -1/8*(7*sqrt(2) *sqrt(-c)*arctan(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)))*tan(b*x + a)^3 - sqrt(2)*sqrt(-c*tan(b *x + a)^2/(tan(b*x + a)^2 - 1))*(9*tan(b*x + a)^2 - 1))/(b*c^2*tan(b*x + a )^3)]
Timed out. \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\int { \frac {\sec \left (2 \, b x + 2 \, a\right )^{3}}{\left (c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx=\int \frac {1}{{\cos \left (2\,a+2\,b\,x\right )}^3\,{\left (c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )\right )}^{3/2}} \,d x \]