Optimal. Leaf size=178 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right )}{2 b c^{3/2}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right )}{4 \sqrt{2} b c^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\sin (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}} \]
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Rubi [A] time = 0.319622, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {4397, 3817, 4022, 3920, 3774, 207, 3795} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right )}{2 b c^{3/2}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right )}{4 \sqrt{2} b c^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\sin (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 3817
Rule 4022
Rule 3920
Rule 3774
Rule 207
Rule 3795
Rubi steps
\begin{align*} \int \frac{\cos (2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx &=\int \frac{\cos (2 a+2 b x)}{(-c+c \sec (2 a+2 b x))^{3/2}} \, dx\\ &=-\frac{\sin (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}-\frac{\int \frac{\cos (2 a+2 b x) \left (3 c+\frac{3}{2} c \sec (2 a+2 b x)\right )}{\sqrt{-c+c \sec (2 a+2 b x)}} \, dx}{2 c^2}\\ &=-\frac{\sin (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{\int \frac{3 c^2+\frac{3}{2} c^2 \sec (2 a+2 b x)}{\sqrt{-c+c \sec (2 a+2 b x)}} \, dx}{2 c^3}\\ &=-\frac{\sin (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{3 \int \sqrt{-c+c \sec (2 a+2 b x)} \, dx}{2 c^2}-\frac{9 \int \frac{\sec (2 a+2 b x)}{\sqrt{-c+c \sec (2 a+2 b x)}} \, dx}{4 c}\\ &=-\frac{\sin (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{3 \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 )}{2 b c}+\frac{9 \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 )}{4 b c}\\ &=-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{-c+c \sec (2 a+2 b x)}}\right )}{2 b c^{3/2}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{-c+c \sec (2 a+2 b x)}}\right )}{4 \sqrt{2} b c^{3/2}}-\frac{\sin (2 a+2 b x)}{4 b (-c+c \sec (2 a+2 b x))^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{-c+c \sec (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 6.19842, size = 342, normalized size = 1.92 \[ \frac{\tan ^2(a+b x) \tan ^2(2 (a+b x)) \left (\frac{1}{2} \sin (2 (a+b x))-\frac{1}{4} \cot (a+b x)-\frac{1}{8} \cot (a+b x) \csc ^2(a+b x)\right )}{b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}-\frac{3 \tan ^{\frac{3}{2}}(a+b x) \tan ^{\frac{3}{2}}(2 (a+b x)) \left (\frac{\tan ^{-1}\left (\sqrt{\tan ^2(a+b x)-1}\right ) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan ^2(a+b x)-1} \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x)}{\left (\tan ^2(a+b x)+1\right )^2}+\frac{\sqrt{2} \cos (2 (a+b x)) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x) \left (2 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right )-\sqrt{2} \tanh ^{-1}\left (\sqrt{1-\tan ^2(a+b x)}\right )\right )}{\sqrt{1-\tan ^2(a+b x)} \left (\tan ^2(a+b x)+1\right )}\right )}{8 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.43, size = 1157, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (2 \, b x + 2 \, a\right )}{\left (c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.91226, size = 1372, normalized size = 7.71 \begin{align*} \left [\frac{9 \, \sqrt{2}{\left (\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}\right )} \sqrt{c} \log \left (\frac{c \tan \left (b x + a\right )^{3} + 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 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3}}\right ) + 12 \,{\left (\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}\right )} \sqrt{c} \log \left (\frac{c \tan \left (b x + a\right )^{3} - 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} - 3 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}\right ) + 2 \, \sqrt{2}{\left (5 \, \tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} - 1\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{16 \,{\left (b c^{2} \tan \left (b x + a\right )^{5} + b c^{2} \tan \left (b x + a\right )^{3}\right )}}, \frac{9 \, \sqrt{2}{\left (\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}\right )} \sqrt{-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{-c}}{c \tan \left (b x + a\right )}\right ) - 12 \,{\left (\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}\right )} \sqrt{-c} \arctan \left (\frac{\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}}{2 \, c \tan \left (b x + a\right )}\right ) + \sqrt{2}{\left (5 \, \tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} - 1\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{8 \,{\left (b c^{2} \tan \left (b x + a\right )^{5} + b c^{2} \tan \left (b x + a\right )^{3}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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