Optimal. Leaf size=36 \[ \frac {\csc (a-c) \log (\cos (b x+c))}{b}-\frac {\csc (a-c) \log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4610, 3475} \[ \frac {\csc (a-c) \log (\cos (b x+c))}{b}-\frac {\csc (a-c) \log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4610
Rubi steps
\begin {align*} \int \sec (a+b x) \sec (c+b x) \, dx &=\csc (a-c) \int \tan (a+b x) \, dx-\csc (a-c) \int \tan (c+b x) \, dx\\ &=-\frac {\csc (a-c) \log (\cos (a+b x))}{b}+\frac {\csc (a-c) \log (\cos (c+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 28, normalized size = 0.78 \[ -\frac {\csc (a-c) (\log (\cos (a+b x))-\log (\cos (b x+c)))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.99, size = 107, normalized size = 2.97 \[ -\frac {\log \left (\cos \left (b x + c\right )^{2}\right ) - \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right )}{2 \, b \sin \left (-a + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 171, normalized size = 4.75 \[ \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1 \right |}\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 55, normalized size = 1.53 \[ -\frac {\ln \left (-\tan \left (b x +a \right ) \cos \relax (a ) \sin \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \cos \relax (c )+\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )}{b \left (\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 349, normalized size = 9.69 \[ -\frac {2 \, {\left ({\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) - \sin \left (2 \, a\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, a\right )\right ) - 2 \, {\left ({\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) - \sin \left (2 \, c\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, c\right )\right ) - {\left ({\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right ) + {\left ({\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right )}{2 \, b \cos \left (2 \, a\right ) \cos \left (2 \, c\right ) - b \cos \left (2 \, c\right )^{2} + 2 \, b \sin \left (2 \, a\right ) \sin \left (2 \, c\right ) - b \sin \left (2 \, c\right )^{2} - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.84, size = 249, normalized size = 6.92 \[ \frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}\,\left (\ln \left (-\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}\,\left (4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )-\ln \left (-\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}\,\left (4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b-b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )} \]
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 \[ \int \sec {\left (a + b x \right )} \sec {\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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