Optimal. Leaf size=129 \[ \frac {\tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b c}+\frac {2 \tan (2 a+2 b x)}{3 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {c \sec (2 a+2 b x)-c}}\right )}{\sqrt {2} b \sqrt {c}} \]
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Rubi [A] time = 0.36, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4397, 3800, 4001, 3795, 207} \[ \frac {\tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b c}+\frac {2 \tan (2 a+2 b x)}{3 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {c \sec (2 a+2 b x)-c}}\right )}{\sqrt {2} b \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 3795
Rule 3800
Rule 4001
Rule 4397
Rubi steps
\begin {align*} \int \frac {\sec ^3(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx &=\int \frac {\sec ^3(2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx\\ &=\frac {\sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b c}+\frac {2 \int \frac {\sec (2 a+2 b x) \left (\frac {c}{2}+c \sec (2 a+2 b x)\right )}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx}{3 c}\\ &=\frac {2 \tan (2 a+2 b x)}{3 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {\sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b c}+\int \frac {\sec (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx\\ &=\frac {2 \tan (2 a+2 b x)}{3 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {\sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b c}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2 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 {\tanh ^{-1}\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 {2 \tan (2 a+2 b x)}{3 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {\sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b c}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 89, normalized size = 0.69 \[ \frac {\cos ^2(a+b x) \csc (2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \left (3 \sqrt {\tan ^2(a+b x)-1} \tan ^{-1}\left (\sqrt {\tan ^2(a+b x)-1}\right )+2 \sec (2 (a+b x))+2\right )}{3 b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.20, size = 294, normalized size = 2.28 \[ \left [\frac {\frac {3 \, \sqrt {2} {\left (c \tan \left (b x + a\right )^{3} - c \tan \left (b x + a\right )\right )} \log \left (\frac {\tan \left (b x + a\right )^{3} - \frac {2 \, \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )}}{\sqrt {c}} - 2 \, \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3}}\right )}{\sqrt {c}} - 8 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{12 \, {\left (b c \tan \left (b x + a\right )^{3} - b c \tan \left (b x + a\right )\right )}}, -\frac {3 \, \sqrt {2} {\left (c \tan \left (b x + a\right )^{3} - c \tan \left (b x + a\right )\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-\frac {1}{c}}}{\tan \left (b x + a\right )}\right ) + 4 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{6 \, {\left (b c \tan \left (b x + a\right )^{3} - b c \tan \left (b x + a\right )\right )}}\right ] \]
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 = 1.21, size = 677, normalized size = 5.25 \[ -\frac {\sqrt {2}\, \left (-1+\cos \left (b x +a \right )\right ) \left (12 \arctanh \left (\frac {\sqrt {4}\, \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-3 \cos \left (b x +a \right )+1\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}}}\, \sin \left (b x +a \right )^{2}}\right ) \left (\cos ^{4}\left (b x +a \right )\right )+12 \ln \left (-\frac {2 \left (\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \left (\cos ^{2}\left (b x +a \right )\right )+\cos \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}}}+1\right )}{\sin \left (b x +a \right )^{2}}\right ) \left (\cos ^{4}\left (b x +a \right )\right )+8 \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\cos ^{4}\left (b x +a \right )\right )+8 \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\cos ^{3}\left (b x +a \right )\right )-12 \arctanh \left (\frac {\sqrt {4}\, \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-3 \cos \left (b x +a \right )+1\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}}}\, \sin \left (b x +a \right )^{2}}\right ) \left (\cos ^{2}\left (b x +a \right )\right )-12 \ln \left (-\frac {2 \left (\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \left (\cos ^{2}\left (b x +a \right )\right )+\cos \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}}}+1\right )}{\sin \left (b x +a \right )^{2}}\right ) \left (\cos ^{2}\left (b x +a \right )\right )+3 \arctanh \left (\frac {\sqrt {4}\, \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-3 \cos \left (b x +a \right )+1\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}}}\, \sin \left (b x +a \right )^{2}}\right )+3 \ln \left (-\frac {2 \left (\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (b x +a \right )\right )-1}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \left (\cos ^{2}\left (b x +a \right )\right )+\cos \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}}}+1\right )}{\sin \left (b x +a \right )^{2}}\right )\right ) \sqrt {4}}{24 b \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-1\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}}}\, \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 ) \left (-3+2 \sqrt {2}\right )^{2} \left (3+2 \sqrt {2}\right )^{2}} \]
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 \[ \int \frac {\sec \left (2 \, b x + 2 \, a\right )^{3}}{\sqrt {c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\cos \left (2\,a+2\,b\,x\right )}^3\,\sqrt {c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )}} \,d x \]
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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