Optimal. Leaf size=110 \[ \frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{5 b}+\frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b} \]
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Rubi [A] time = 0.27, antiderivative size = 110, normalized size of antiderivative = 1.00, 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, 3798, 3793, 3792} \[ \frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{5 b}+\frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 3798
Rule 4397
Rubi steps
\begin {align*} \int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx &=\int \sec ^2(2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx\\ &=\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}-\frac {3}{5} \int \sec (2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx\\ &=-\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}+\frac {1}{5} (4 c) \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 59, normalized size = 0.54 \[ \frac {\cot (a+b x) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} (4 \cot (a+b x) \cot (2 (a+b x))+\sec (2 (a+b x))-2)}{5 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 88, normalized size = 0.80 \[ \frac {2 \, \sqrt {2} {\left (5 \, c \tan \left (b x + a\right )^{4} - 5 \, c \tan \left (b x + a\right )^{2} + 2 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{5 \, {\left (b \tan \left (b x + a\right )^{5} - 2 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}} \]
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 [A] time = 0.86, size = 85, normalized size = 0.77 \[ \frac {2 \sqrt {2}\, \left (12 \left (\cos ^{4}\left (b x +a \right )\right )-15 \left (\cos ^{2}\left (b x +a \right )\right )+5\right ) \cos \left (b x +a \right ) \left (\frac {c \left (\sin ^{2}\left (b x +a \right )\right )}{2 \left (\cos ^{2}\left (b x +a \right )\right )-1}\right )^{\frac {3}{2}}}{5 b \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-1\right ) \sin \left (b x +a \right )^{3}} \]
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 [B] time = 11.04, size = 149, normalized size = 1.35 \[ \frac {2\,c\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{5\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}^2} \]
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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