Optimal. Leaf size=100 \[ \frac {2 \tan ^3(x)}{3 (\tan (x) \tan (2 x))^{3/2}}+\frac {3 \tan (x)}{4 \sqrt {\tan (x) \tan (2 x)}}+\frac {\tan (x)}{2 (\tan (x) \tan (2 x))^{3/2}}+2 \tanh ^{-1}\left (\frac {\tan (x)}{\sqrt {\tan (x) \tan (2 x)}}\right )-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {2} \tan (x)}{\sqrt {\tan (x) \tan (2 x)}}\right )}{4 \sqrt {2}} \]
________________________________________________________________________________________
Rubi [B] time = 1.23, antiderivative size = 208, normalized size of antiderivative = 2.08, number of steps used = 21, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {4397, 12, 6719, 6725, 266, 47, 50, 63, 203, 444} \[ \frac {\left (1-\tan ^2(x)\right ) \tan (x)}{3 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {3 \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {11 \tan ^{-1}\left (\sqrt {\tan ^2(x)-1}\right ) \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {\tan ^2(x)-1}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\tan ^2(x)-1}}{\sqrt {2}}\right ) \tan (x)}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {\tan ^2(x)-1}}+\frac {\left (1-\tan ^2(x)\right ) \cot (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 47
Rule 50
Rule 63
Rule 203
Rule 266
Rule 444
Rule 4397
Rule 6719
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx &=\int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(-1+\sec (2 x))^{3/2}} \, dx\\ &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+3 x^2+2 x^4\right )}{2 \sqrt {2} x^2 \sqrt {\frac {x^2}{1-x^2}} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+3 x^2+2 x^4\right )}{x^2 \sqrt {\frac {x^2}{1-x^2}} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{2 \sqrt {2}}\\ &=\frac {\tan (x) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (-1+3 x^2+2 x^4\right )}{x^3 \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=\frac {\tan (x) \operatorname {Subst}\left (\int \left (-\frac {\left (1-x^2\right )^{3/2}}{x^3}+\frac {4 \left (1-x^2\right )^{3/2}}{x}-\frac {2 x \left (1-x^2\right )^{3/2}}{1+x^2}\right ) \, dx,x,\tan (x)\right )}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=-\frac {\tan (x) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{x^3} \, dx,x,\tan (x)\right )}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\tan (x) \operatorname {Subst}\left (\int \frac {x \left (1-x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\left (\sqrt {2} \tan (x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{x} \, dx,x,\tan (x)\right )}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=-\frac {\tan (x) \operatorname {Subst}\left (\int \frac {(1-x)^{3/2}}{x^2} \, dx,x,\tan ^2(x)\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\tan (x) \operatorname {Subst}\left (\int \frac {(1-x)^{3/2}}{1+x} \, dx,x,\tan ^2(x)\right )}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\tan (x) \operatorname {Subst}\left (\int \frac {(1-x)^{3/2}}{x} \, dx,x,\tan ^2(x)\right )}{\sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=\frac {\cot (x) \left (1-\tan ^2(x)\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {\tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\sqrt {2} \tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {(3 \tan (x)) \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,\tan ^2(x)\right )}{8 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\tan (x) \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,\tan ^2(x)\right )}{\sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\tan (x) \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{1+x} \, dx,x,\tan ^2(x)\right )}{\sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=\frac {3 \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\cot (x) \left (1-\tan ^2(x)\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {\tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\sqrt {2} \tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {(3 \tan (x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\tan ^2(x)\right )}{8 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\tan (x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\tan ^2(x)\right )}{\sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\left (\sqrt {2} \tan (x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)} \, dx,x,\tan ^2(x)\right )}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=\frac {3 \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\cot (x) \left (1-\tan ^2(x)\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {\tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\sqrt {2} \tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {(3 \tan (x)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\tan ^2(x)}\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\left (\sqrt {2} \tan (x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\tan ^2(x)}\right )}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\left (2 \sqrt {2} \tan (x)\right ) \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-\tan ^2(x)}\right )}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\\ &=\frac {3 \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\tan ^2(x)}\right ) \tan (x)}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}-\frac {\sqrt {2} \tanh ^{-1}\left (\sqrt {1-\tan ^2(x)}\right ) \tan (x)}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-\tan ^2(x)}}{\sqrt {2}}\right ) \tan (x)}{\sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}+\frac {\cot (x) \left (1-\tan ^2(x)\right )}{4 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}-\frac {\tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}+\frac {\sqrt {2} \tan (x) \left (1-\tan ^2(x)\right )}{3 \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.83, size = 168, normalized size = 1.68 \[ \frac {\tan ^2(2 x) \left (2 \tan ^2(x)-\cos (2 x)\right ) \left (-3 \sin (x) \cos (x) \tan ^{-1}\left (\sqrt {\tan ^2(x)-1}\right ) \sqrt {\tan ^2(x)-1}+\frac {4 \sqrt {2} \cos (2 x) \tan (x) \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {1-\tan ^2(x)}\right )-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-2 \tan ^2(x)}\right )\right )}{\sqrt {1-\tan ^2(x)}}+\frac {1}{3} \left (-3 \cot (x)-4 \sin (x) \cos (x)+(9 \cos (2 x)+5) \tan ^3(x)\right )\right )}{2 (6 \cos (2 x)+\cos (4 x)-3) (\tan (x) \tan (2 x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.49, size = 271, normalized size = 2.71 \[ -\frac {24 \, {\left (\cos \relax (x)^{5} - \cos \relax (x)^{3}\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (8 \, \cos \relax (x)^{5} - 6 \, \cos \relax (x)^{3} + \cos \relax (x)\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{2 \, \cos \relax (x)^{2} - 1}} - {\left (32 \, \cos \relax (x)^{4} - 16 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)}{\sin \relax (x)}\right ) \sin \relax (x) - 33 \, {\left (\sqrt {2} \cos \relax (x)^{5} - \sqrt {2} \cos \relax (x)^{3}\right )} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (2 \, {\left (3 \, \sqrt {2} - 4\right )} \cos \relax (x)^{3} - {\left (3 \, \sqrt {2} - 4\right )} \cos \relax (x)\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{2 \, \cos \relax (x)^{2} - 1}} + {\left (3 \, {\left (2 \, \sqrt {2} - 3\right )} \cos \relax (x)^{2} - 2 \, \sqrt {2} + 3\right )} \sin \relax (x)\right )}}{{\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}\right ) \sin \relax (x) - 2 \, \sqrt {2} {\left (22 \, \cos \relax (x)^{6} - 47 \, \cos \relax (x)^{4} + 26 \, \cos \relax (x)^{2} - 4\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{2 \, \cos \relax (x)^{2} - 1}} - 44 \, {\left (\cos \relax (x)^{5} - \cos \relax (x)^{3}\right )} \sin \relax (x)}{48 \, {\left (\cos \relax (x)^{5} - \cos \relax (x)^{3}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.87, size = 193, normalized size = 1.93 \[ -\frac {\sqrt {2} {\left (2 \, {\left (-\tan \relax (x)^{2} + 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {-\tan \relax (x)^{2} + 1}\right )}}{12 \, \mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right )} + \frac {11 \, \sqrt {2} \log \left (\sqrt {-\tan \relax (x)^{2} + 1} + 1\right )}{16 \, \mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right )} - \frac {11 \, \sqrt {2} \log \left (-\sqrt {-\tan \relax (x)^{2} + 1} + 1\right )}{16 \, \mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right )} + \frac {\log \left (\frac {\sqrt {2} - \sqrt {-\tan \relax (x)^{2} + 1}}{\sqrt {2} + \sqrt {-\tan \relax (x)^{2} + 1}}\right )}{\mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right )} - \frac {\sqrt {2} \sqrt {-\tan \relax (x)^{2} + 1}}{8 \, \mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right ) \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.02, size = 559, normalized size = 5.59
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {4}\, \left (-1+\cos \relax (x )\right )^{2} \left (-48 \left (\cos ^{4}\relax (x )\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \cos \relax (x ) \sqrt {4}\, \left (-1+\cos \relax (x )\right )}{2 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \sin \relax (x )^{2}}\right )+22 \left (\cos ^{4}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}-168 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {4 \left (\left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}-2 \left (\cos ^{2}\relax (x )\right )+\cos \relax (x )-\sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}+1\right )}{\sin \relax (x )^{2}}\right )+33 \left (\cos ^{4}\relax (x )\right ) \arctanh \left (\frac {\sqrt {4}\, \left (2 \left (\cos ^{2}\relax (x )\right )-3 \cos \relax (x )+1\right )}{2 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \sin \relax (x )^{2}}\right )+201 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {2 \left (\left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}-2 \left (\cos ^{2}\relax (x )\right )+\cos \relax (x )-\sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}+1\right )}{\sin \relax (x )^{2}}\right )+48 \left (\cos ^{3}\relax (x )\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \cos \relax (x ) \sqrt {4}\, \left (-1+\cos \relax (x )\right )}{2 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \sin \relax (x )^{2}}\right )+168 \left (\cos ^{3}\relax (x )\right ) \ln \left (-\frac {4 \left (\left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}-2 \left (\cos ^{2}\relax (x )\right )+\cos \relax (x )-\sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}+1\right )}{\sin \relax (x )^{2}}\right )-33 \left (\cos ^{3}\relax (x )\right ) \arctanh \left (\frac {\sqrt {4}\, \left (2 \left (\cos ^{2}\relax (x )\right )-3 \cos \relax (x )+1\right )}{2 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \sin \relax (x )^{2}}\right )-201 \left (\cos ^{3}\relax (x )\right ) \ln \left (-\frac {2 \left (\left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}-2 \left (\cos ^{2}\relax (x )\right )+\cos \relax (x )-\sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}+1\right )}{\sin \relax (x )^{2}}\right )-36 \left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}+8 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\right )}{96 \cos \relax (x )^{3} \sin \relax (x )^{3} \left (\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}\right )^{\frac {3}{2}} \left (\frac {\sin ^{2}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right )-1}\right )^{\frac {3}{2}}}\) | \(559\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, \tan \relax (x)^{2} - \cos \left (2 \, x\right )}{\left (\tan \left (2 \, x\right ) \tan \relax (x)\right )^{\frac {3}{2}} \cos \relax (x)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\cos \left (2\,x\right )-2\,{\mathrm {tan}\relax (x)}^2}{{\cos \relax (x)}^2\,{\left (\mathrm {tan}\left (2\,x\right )\,\mathrm {tan}\relax (x)\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________