Optimal. Leaf size=110 \[ \frac{\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b c}+\frac{2 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{15 b}+\frac{7 c \tan (2 a+2 b x)}{15 b \sqrt{c \sec (2 a+2 b x)-c}} \]
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Rubi [A] time = 0.2762, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {4397, 3800, 4001, 3792} \[ \frac{\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b c}+\frac{2 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{15 b}+\frac{7 c \tan (2 a+2 b x)}{15 b \sqrt{c \sec (2 a+2 b x)-c}} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx &=\int \sec ^3(2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac{(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b c}+\frac{2 \int \sec (2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \left (\frac{3 c}{2}+c \sec (2 a+2 b x)\right ) \, dx}{5 c}\\ &=\frac{2 \sqrt{-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{15 b}+\frac{(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b c}+\frac{7}{15} \int \sec (2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac{7 c \tan (2 a+2 b x)}{15 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{2 \sqrt{-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{15 b}+\frac{(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b c}\\ \end{align*}
Mathematica [A] time = 0.18082, size = 62, normalized size = 0.56 \[ \frac{(5 \cos (a+b x)+2 \cos (5 (a+b x))) \csc (a+b x) \sec ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}{15 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.466, size = 88, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{4}\cos \left ( bx+a \right ) \left ( 32\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-40\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+15 \right ) }{30\,b\sin \left ( bx+a \right ) \left ( 2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-1 \right ) ^{2}}\sqrt{{\frac{c \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09083, size = 216, normalized size = 1.96 \begin{align*} \frac{\sqrt{2}{\left (15 \, \tan \left (b x + a\right )^{4} - 10 \, \tan \left (b x + a\right )^{2} + 7\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{15 \,{\left (b \tan \left (b x + a\right )^{5} - 2 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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