Optimal. Leaf size=89 \[ -\cos (x)+\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {1}{6} \sqrt {2-\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{6} \sqrt {2+\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {12, 6742, 2073, 207, 1166} \[ -\cos (x)+\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {1}{6} \sqrt {2-\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{6} \sqrt {2+\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 1166
Rule 2073
Rule 6742
Rubi steps
\begin {align*} \int \cos (x) \tan (6 x) \, dx &=-\operatorname {Subst}\left (\int \frac {2 x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {1-12 x^2+16 x^4}{2 \left (1-18 x^2+48 x^4-32 x^6\right )}\right ) \, dx,x,\cos (x)\right )\right )\\ &=-\cos (x)+\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\\ &=-\cos (x)+\operatorname {Subst}\left (\int \left (-\frac {1}{3 \left (-1+2 x^2\right )}-\frac {2 \left (-1+8 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cos (x)\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {-1+8 x^2}{1-16 x^2+16 x^4} \, dx,x,\cos (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}-\cos (x)-\frac {1}{3} \left (4 \left (2-\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-8+4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right )-\frac {1}{3} \left (4 \left (2+\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-8-4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {1}{6} \sqrt {2-\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{6} \sqrt {2+\sqrt {3}} \tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )-\cos (x)\\ \end {align*}
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Mathematica [C] time = 9.04, size = 679, normalized size = 7.63 \[ -\cos (x)+\left (-\frac {1}{6}-\frac {i}{6}\right ) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \sec \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )+\left (\frac {1}{6}+\frac {i}{6}\right ) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )\right )-\frac {\left (1+\sqrt {2}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )-2 \sqrt {3} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+2}{\sqrt {6}}\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (2 \sin (x)-2 \cos (x)+\sqrt {2}\right )\right )\right )}{12 \left (2+\sqrt {2}\right )}+\frac {x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\left (\sqrt {2}-1\right ) \tan \left (\frac {x}{2}\right )+\sqrt {2}}{\sqrt {3}}\right )+\log \left (\sec ^2\left (\frac {x}{2}\right ) \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+1\right )\right )}{12 \sqrt {2}}-\frac {\left (\sqrt {2}-\sqrt {3} \sin (x)\right ) \left (\left (\sqrt {6}-2\right ) \sin (x)-\left (\left (\sqrt {6}-2\right ) \cos (x)\right )+\sqrt {6}-3\right ) \left (2 \left (\sqrt {6}-2\right ) \tanh ^{-1}\left (\left (\sqrt {2}-\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )+\sqrt {2}\right )+\left (3 \sqrt {2}-2 \sqrt {3}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+\sqrt {3}\right )\right )\right )\right )}{12 \left (20 \sqrt {6} \sin (x)-50 \sin (x)-5 \sqrt {6} \sin (2 x)+12 \sin (2 x)+\left (20-8 \sqrt {6}\right ) \cos (x)+\left (12-5 \sqrt {6}\right ) \cos (2 x)+15 \sqrt {6}-36\right )}+\frac {\left (\sqrt {6} \sin (x)+2\right ) \left (\left (2+\sqrt {6}\right ) \sin (x)-\left (\left (2+\sqrt {6}\right ) \cos (x)\right )+\sqrt {6}+3\right ) \left (\left (3+\sqrt {6}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (2 \sin (x)-2 \cos (x)+\sqrt {6}\right )\right )\right )-2 \left (\sqrt {2}+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (2+\sqrt {6}\right ) \tan \left (\frac {x}{2}\right )+2}{\sqrt {2}}\right )\right )}{12 \left (-20 \sqrt {6} \sin (x)-50 \sin (x)+5 \sqrt {6} \sin (2 x)+12 \sin (2 x)+4 \left (5+2 \sqrt {6}\right ) \cos (x)+\left (12+5 \sqrt {6}\right ) \cos (2 x)-15 \sqrt {6}-36\right )} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.85, size = 134, normalized size = 1.51 \[ \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} + 2 \, \cos \relax (x)\right ) - \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} - 2 \, \cos \relax (x)\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\sqrt {-\sqrt {3} + 2} + 2 \, \cos \relax (x)\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\sqrt {-\sqrt {3} + 2} - 2 \, \cos \relax (x)\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 1}{2 \, \cos \relax (x)^{2} - 1}\right ) - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \relax (x) \tan \left (6 \, x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 104, normalized size = 1.17 \[ -\cos \relax (x )+\frac {2 \left (-3+2 \sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {8 \cos \relax (x )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{9 \left (2 \sqrt {6}-2 \sqrt {2}\right )}+\frac {2 \left (3+2 \sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {8 \cos \relax (x )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{9 \left (2 \sqrt {6}+2 \sqrt {2}\right )}+\frac {\arctanh \left (\cos \relax (x ) \sqrt {2}\right ) \sqrt {2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{24} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) + 2 \, {\left (\sqrt {2} \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 1\right ) - \frac {1}{24} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) - 2 \, {\left (\sqrt {2} \cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 1\right ) - \cos \relax (x) - \int \frac {{\left (2 \, \sin \left (7 \, x\right ) + \sin \left (5 \, x\right ) - \sin \left (3 \, x\right ) - 2 \, \sin \relax (x)\right )} \cos \left (8 \, x\right ) + {\left (\sin \left (3 \, x\right ) + 2 \, \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (2 \, \cos \left (7 \, x\right ) + \cos \left (5 \, x\right ) - \cos \left (3 \, x\right ) - 2 \, \cos \relax (x)\right )} \sin \left (8 \, x\right ) - 2 \, {\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (7 \, x\right ) - {\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) - {\left (\cos \left (3 \, x\right ) + 2 \, \cos \relax (x)\right )} \sin \left (4 \, x\right ) + 2 \, \cos \left (7 \, x\right ) \sin \left (4 \, x\right ) + \cos \left (5 \, x\right ) \sin \left (4 \, x\right ) - \sin \left (3 \, x\right ) - 2 \, \sin \relax (x)}{3 \, {\left (2 \, {\left (\cos \left (4 \, x\right ) - 1\right )} \cos \left (8 \, x\right ) - \cos \left (8 \, x\right )^{2} - \cos \left (4 \, x\right )^{2} - \sin \left (8 \, x\right )^{2} + 2 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) - \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 787, normalized size = 8.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \tan {\left (6 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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