Integrand size = 31, antiderivative size = 88 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {-c+c \sec (2 a+2 b x)}}\right )}{\sqrt {2} b \sqrt {c}}+\frac {\tan (2 a+2 b x)}{b \sqrt {-c+c \sec (2 a+2 b x)}} \]
-1/2*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*2^(1/2)/c^(1/2)+tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)
Time = 0.74 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\frac {\left (2+\arctan \left (\sqrt {-1+\tan ^2(a+b x)}\right ) \sqrt {-1+\tan ^2(a+b x)}\right ) \tan (2 (a+b x))}{2 b \sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \]
((2 + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*( a + b*x)])/(2*b*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])
Time = 0.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 4897, 3042, 4285, 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.
\(\displaystyle \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (2 a+2 b x)^2}{\sqrt {c \tan (a+b x) \tan (2 a+2 b x)}}dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int \frac {\sec ^2(2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (2 a+2 b x+\frac {\pi }{2}\right )^2}{\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}dx\) |
\(\Big \downarrow \) 4285 |
\(\displaystyle \int \frac {\sec (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}dx+\frac {\tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (2 a+2 b x+\frac {\pi }{2}\right )}{\sqrt {c \csc \left (2 a+2 b x+\frac {\pi }{2}\right )-c}}dx+\frac {\tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}\) |
\(\Big \downarrow \) 4282 |
\(\displaystyle \frac {\tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {\int \frac {1}{\frac {c^2 \tan ^2(2 a+2 b x)}{c \sec (2 a+2 b x)-c}-2 c}d\left (-\frac {c \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{b}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {\tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {c \sec (2 a+2 b x)-c}}\right )}{\sqrt {2} b \sqrt {c}}\) |
-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x] ])]/(Sqrt[2]*b*Sqrt[c])) + Tan[2*a + 2*b*x]/(b*Sqrt[-c + c*Sec[2*a + 2*b*x ]])
3.7.21.3.1 Defintions of rubi rules used
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_)]^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)]
Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(77)=154\).
Time = 4.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.20
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\cos \left (x b +a \right )-1\right ) \left (\operatorname {arctanh}\left (\frac {2 \cos \left (x b +a \right )-1}{\left (1+\cos \left (x b +a \right )\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}}\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}\, \sin \left (x b +a \right )^{2}-\ln \left (\frac {2 \cos \left (x b +a \right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}+2 \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}-4 \cos \left (x b +a \right )-2}{1+\cos \left (x b +a \right )}\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}\, \sin \left (x b +a \right )^{2}-2 \sin \left (x b +a \right )^{2}+2 \cos \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )+2\right ) \sin \left (x b +a \right ) \sqrt {4}}{2 b \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \left (\sin \left (x b +a \right )^{4}-6 \cos \left (x b +a \right )^{2} \sin \left (x b +a \right )^{2}+\cos \left (x b +a \right )^{4}+12 \cos \left (x b +a \right ) \sin \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )^{3}-6 \sin \left (x b +a \right )^{2}+6 \cos \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )+1\right )}\) | \(370\) |
1/2*2^(1/2)/b*(cos(b*x+a)-1)*(arctanh((2*cos(b*x+a)-1)/(1+cos(b*x+a))/((2* cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a) )^2)^(1/2)*sin(b*x+a)^2-ln(2*(cos(b*x+a)*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a) )^2)^(1/2)+((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)-2*cos(b*x+a)-1)/(1+ cos(b*x+a)))*((2*cos(b*x+a)^2-1)/(1+cos(b*x+a))^2)^(1/2)*sin(b*x+a)^2-2*si n(b*x+a)^2+2*cos(b*x+a)^2-4*cos(b*x+a)+2)*sin(b*x+a)/(c*sin(b*x+a)^2/(2*co s(b*x+a)^2-1))^(1/2)/(sin(b*x+a)^4-6*cos(b*x+a)^2*sin(b*x+a)^2+cos(b*x+a)^ 4+12*cos(b*x+a)*sin(b*x+a)^2-4*cos(b*x+a)^3-6*sin(b*x+a)^2+6*cos(b*x+a)^2- 4*cos(b*x+a)+1)*4^(1/2)
Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.78 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\left [\frac {\sqrt {2} \sqrt {c} \log \left (\frac {\tan \left (b x + a\right )^{3} - \frac {2 \, \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )}}{\sqrt {c}} - 2 \, \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3}}\right ) \tan \left (b x + a\right ) + 4 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{4 \, b c \tan \left (b x + a\right )}, -\frac {\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-\frac {1}{c}}}{\tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - 2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{2 \, b c \tan \left (b x + a\right )}\right ] \]
[1/4*(sqrt(2)*sqrt(c)*log((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*tan(b*x + a))/tan(b*x + a)^3)*tan(b*x + a) + 4*sqrt(2)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1) ))/(b*c*tan(b*x + a)), -1/2*(sqrt(2)*c*sqrt(-1/c)*arctan(sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1))*(tan(b*x + a)^2 - 1)*sqrt(-1/c)/tan(b*x + a))* tan(b*x + a) - 2*sqrt(2)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)))/(b* c*tan(b*x + a))]
Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\int { \frac {\sec \left (2 \, b x + 2 \, a\right )^{2}}{\sqrt {c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\int \frac {1}{{\cos \left (2\,a+2\,b\,x\right )}^2\,\sqrt {c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )}} \,d x \]