Optimal. Leaf size=148 \[ -\frac {76 c^2 \tan (2 a+2 b x)}{105 b \sqrt {c \sec (2 a+2 b x)-c}}+\frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{5/2}}{7 b c}+\frac {2 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{35 b}+\frac {19 c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{105 b} \]
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Rubi [A] time = 0.35, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4397, 3800, 4001, 3793, 3792} \[ -\frac {76 c^2 \tan (2 a+2 b x)}{105 b \sqrt {c \sec (2 a+2 b x)-c}}+\frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{5/2}}{7 b c}+\frac {2 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{35 b}+\frac {19 c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{105 b} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 3800
Rule 4001
Rule 4397
Rubi steps
\begin {align*} \int \sec ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx &=\int \sec ^3(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))^{5/2} \tan (2 a+2 b x)}{7 b c}+\frac {2 \int \sec (2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \left (\frac {5 c}{2}+c \sec (2 a+2 b x)\right ) \, dx}{7 c}\\ &=\frac {2 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b}+\frac {(-c+c \sec (2 a+2 b x))^{5/2} \tan (2 a+2 b x)}{7 b c}+\frac {19}{35} \int \sec (2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx\\ &=\frac {19 c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{105 b}+\frac {2 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b}+\frac {(-c+c \sec (2 a+2 b x))^{5/2} \tan (2 a+2 b x)}{7 b c}-\frac {1}{105} (76 c) \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac {76 c^2 \tan (2 a+2 b x)}{105 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {19 c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{105 b}+\frac {2 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b}+\frac {(-c+c \sec (2 a+2 b x))^{5/2} \tan (2 a+2 b x)}{7 b c}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 73, normalized size = 0.49 \[ -\frac {\cot (a+b x) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \left (76 \cot (a+b x) \cot (2 (a+b x))-15 \sec ^2(2 (a+b x))+24 \sec (2 (a+b x))-28\right )}{105 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 111, normalized size = 0.75 \[ -\frac {2 \, \sqrt {2} {\left (105 \, c \tan \left (b x + a\right )^{6} - 140 \, c \tan \left (b x + a\right )^{4} + 133 \, c \tan \left (b x + a\right )^{2} - 38 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{105 \, {\left (b \tan \left (b x + a\right )^{7} - 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} - b \tan \left (b x + a\right )\right )}} \]
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 = 0.94, size = 95, normalized size = 0.64 \[ -\frac {2 \sqrt {2}\, \left (416 \left (\cos ^{6}\left (b x +a \right )\right )-728 \left (\cos ^{4}\left (b x +a \right )\right )+455 \left (\cos ^{2}\left (b x +a \right )\right )-105\right ) \cos \left (b x +a \right ) \left (\frac {c \left (\sin ^{2}\left (b x +a \right )\right )}{2 \left (\cos ^{2}\left (b x +a \right )\right )-1}\right )^{\frac {3}{2}}}{105 b \left (2 \left (\cos ^{2}\left (b x +a \right )\right )-1\right )^{2} \sin \left (b x +a \right )^{3}} \]
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 = 9.26, size = 479, normalized size = 3.24 \[ \frac {\left (\frac {c\,8{}\mathrm {i}}{7\,b}-\frac {c\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,8{}\mathrm {i}}{7\,b}\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 )}}}{\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 )}^3}-\frac {\left (\frac {c\,4{}\mathrm {i}}{5\,b}-\frac {c\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,92{}\mathrm {i}}{35\,b}\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 )}}}{\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}-\frac {\left (\frac {c\,4{}\mathrm {i}}{3\,b}+\frac {c\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,52{}\mathrm {i}}{105\,b}\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 )}}}{\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 )}-\frac {c\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\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 )}}\,104{}\mathrm {i}}{105\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )} \]
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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