Optimal. Leaf size=45 \[ -\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4397, 3774, 207} \[ -\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 3774
Rule 4397
Rubi steps
\begin {align*} \int \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx &=\int \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {1}{-c+x^2} \, dx,x,-\frac {c \tan (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}}\right )}{b}\\ &=-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 73, normalized size = 1.62 \[ -\frac {\sqrt {\cos (2 (a+b x))} \csc (a+b x) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \tanh ^{-1}\left (\frac {\sqrt {2} \cos (a+b x)}{\sqrt {\cos (2 (a+b x))}}\right )}{\sqrt {2} b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 201, normalized size = 4.47 \[ \left [\frac {\sqrt {c} \log \left (-\frac {c \tan \left (b x + a\right )^{5} - 14 \, c \tan \left (b x + a\right )^{3} - 4 \, \sqrt {2} {\left (\tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} + 3\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} \sqrt {c} + 17 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{5} + 2 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}\right )}{4 \, b}, \frac {\sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-c}}{c \tan \left (b x + a\right )^{3} - 3 \, c \tan \left (b x + a\right )}\right )}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.94, size = 136, normalized size = 3.02 \[ -\frac {\sqrt {\frac {c \left (1-\left (\cos ^{2}\left (b x +a \right )\right )\right )}{2 \left (\cos ^{2}\left (b x +a \right )\right )-1}}\, \sin \left (b x +a \right ) \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \arctanh \left (\frac {\cos \left (b x +a \right ) \sqrt {4}\, \left (-1+\cos \left (b x +a \right )\right ) \sqrt {2}}{2 \sin \left (b x +a \right )^{2} \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\right ) \sqrt {4}}{2 b \left (-1+\cos \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 430, normalized size = 9.56 \[ \frac {\sqrt {c} {\left (\log \left (4 \, \sqrt {\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right )^{2} + 4 \, \sqrt {\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right )^{2} + 8 \, {\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right ) + 4\right ) - \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + \sqrt {\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right )^{2}\right )} + 2 \, {\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (2 \, b x + 2 \, a\right ) \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right ) + \sin \left (2 \, b x + 2 \, a\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, b x + 4 \, a\right ), \cos \left (4 \, b x + 4 \, a\right ) + 1\right )\right )\right )}\right )\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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