Optimal. Leaf size=182 \[ -\frac {11 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{16 b}+\frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {c \sec (2 a+2 b x)-c}}+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {11 c^2 \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt {c \sec (2 a+2 b x)-c}} \]
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Rubi [A] time = 0.31, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4397, 3813, 21, 3805, 3774, 207} \[ \frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {c \sec (2 a+2 b x)-c}}+\frac {c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {11 c^2 \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {11 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {c \sec (2 a+2 b x)-c}}\right )}{16 b} \]
Antiderivative was successfully verified.
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Rule 21
Rule 207
Rule 3774
Rule 3805
Rule 3813
Rule 4397
Rubi steps
\begin {align*} \int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx &=\int \cos ^3(2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx\\ &=\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {1}{3} c \int \frac {\cos ^2(2 a+2 b x) \left (\frac {11 c}{2}-\frac {11}{2} c \sec (2 a+2 b x)\right )}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx\\ &=\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {1}{6} (11 c) \int \cos ^2(2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac {11 c^2 \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {1}{8} (11 c) \int \cos (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {11 c^2 \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {1}{16} (11 c) \int \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {11 c^2 \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {\left (11 c^2\right ) \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 )}{16 b}\\ &=-\frac {11 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}}\right )}{16 b}+\frac {11 c^2 \sin (2 a+2 b x)}{16 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {11 c^2 \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c^2 \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt {-c+c \sec (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 117, normalized size = 0.64 \[ \frac {c \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \left (-42 \sin (2 (a+b x))+14 \sin (4 (a+b x))-4 \sin (6 (a+b x))+38 \cot (a+b x)-33 \sqrt {2} \sqrt {\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left (\frac {\sqrt {2} \cos (a+b x)}{\sqrt {\cos (2 (a+b x))}}\right )\right )}{96 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 503, normalized size = 2.76 \[ \left [\frac {33 \, {\left (c \tan \left (b x + a\right )^{7} + 3 \, c \tan \left (b x + a\right )^{5} + 3 \, c \tan \left (b x + a\right )^{3} + c \tan \left (b x + a\right )\right )} \sqrt {c} \log \left (-\frac {c \tan \left (b x + a\right )^{5} - 14 \, c \tan \left (b x + a\right )^{3} - 4 \, \sqrt {2} {\left (\tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} + 3\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} \sqrt {c} + 17 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{5} + 2 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}\right ) - 4 \, \sqrt {2} {\left (63 \, c \tan \left (b x + a\right )^{6} - 13 \, c \tan \left (b x + a\right )^{4} - 31 \, c \tan \left (b x + a\right )^{2} - 19 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{192 \, {\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}, \frac {33 \, {\left (c \tan \left (b x + a\right )^{7} + 3 \, c \tan \left (b x + a\right )^{5} + 3 \, c \tan \left (b x + a\right )^{3} + c \tan \left (b x + a\right )\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-c}}{c \tan \left (b x + a\right )^{3} - 3 \, c \tan \left (b x + a\right )}\right ) - 2 \, \sqrt {2} {\left (63 \, c \tan \left (b x + a\right )^{6} - 13 \, c \tan \left (b x + a\right )^{4} - 31 \, c \tan \left (b x + a\right )^{2} - 19 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{96 \, {\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}\right ] \]
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 = 1.11, size = 1078, normalized size = 5.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (2\,a+2\,b\,x\right )}^3\,{\left (c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )\right )}^{3/2} \,d x \]
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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