Optimal. Leaf size=43 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{16 b}+\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4373, 2702,
308, 213} \begin {gather*} \frac {\sec ^3(a+b x)}{48 b}+\frac {\sec (a+b x)}{16 b}-\frac {\tanh ^{-1}(\cos (a+b x))}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 308
Rule 2702
Rule 4373
Rubi steps
\begin {align*} \int \csc ^4(2 a+2 b x) \sin ^3(a+b x) \, dx &=\frac {1}{16} \int \csc (a+b x) \sec ^4(a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=\frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{16 b}+\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 1.42 \begin {gather*} \frac {1}{16} \left (-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 41, normalized size = 0.95
method | result | size |
default | \(\frac {\frac {1}{3 \cos \left (x b +a \right )^{3}}+\frac {1}{\cos \left (x b +a \right )}+\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16 b}\) | \(41\) |
risch | \(\frac {3 \,{\mathrm e}^{5 i \left (x b +a \right )}+10 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{16 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{16 b}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 987 vs.
\(2 (37) = 74\).
time = 0.32, size = 987, normalized size = 22.95 \begin {gather*} \frac {4 \, {\left (3 \, \cos \left (5 \, b x + 5 \, a\right ) + 10 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \cos \left (6 \, b x + 6 \, a\right ) + 12 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (5 \, b x + 5 \, a\right ) + 12 \, {\left (10 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) + 40 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (3 \, b x + 3 \, a\right ) + 36 \, \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) - 3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (6 \, b x + 6 \, a\right ) + \cos \left (6 \, b x + 6 \, a\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 9 \, \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 6 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 6 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (6 \, b x + 6 \, a\right ) + \cos \left (6 \, b x + 6 \, a\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 9 \, \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 6 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 6 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 4 \, {\left (3 \, \sin \left (5 \, b x + 5 \, a\right ) + 10 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + 36 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (5 \, b x + 5 \, a\right ) + 12 \, {\left (10 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \sin \left (4 \, b x + 4 \, a\right ) + 120 \, \sin \left (3 \, b x + 3 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 36 \, \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 12 \, \cos \left (b x + a\right )}{96 \, {\left (b \cos \left (6 \, b x + 6 \, a\right )^{2} + 9 \, b \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, b \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, b \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (3 \, b \cos \left (4 \, b x + 4 \, a\right ) + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (6 \, b x + 6 \, a\right ) + 6 \, {\left (3 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (4 \, b x + 4 \, a\right ) + 6 \, b \cos \left (2 \, b x + 2 \, a\right ) + 6 \, {\left (b \sin \left (4 \, b x + 4 \, a\right ) + b \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.86, size = 67, normalized size = 1.56 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{3} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, \cos \left (b x + a\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (b x + a\right )^{2} - 2}{96 \, b \cos \left (b x + a\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (37) = 74\).
time = 0.47, size = 98, normalized size = 2.28 \begin {gather*} \frac {\frac {8 \, {\left (\frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 2\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 3 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{96 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 37, normalized size = 0.86 \begin {gather*} \frac {\frac {{\cos \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,{\cos \left (a+b\,x\right )}^3}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{16\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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