Optimal. Leaf size=43 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b} \]
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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 = {4372, 2701,
308, 213} \begin {gather*} -\frac {\csc ^3(a+b x)}{48 b}-\frac {\csc (a+b x)}{16 b}+\frac {\tanh ^{-1}(\sin (a+b x))}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 308
Rule 2701
Rule 4372
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac {1}{16} \int \csc ^4(a+b x) \sec (a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 31, normalized size = 0.72 \begin {gather*} -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\sin ^2(a+b x)\right )}{48 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 41, normalized size = 0.95
method | result | size |
default | \(\frac {-\frac {1}{3 \sin \left (x b +a \right )^{3}}-\frac {1}{\sin \left (x b +a \right )}+\ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{16 b}\) | \(41\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i \left (x b +a \right )}-10 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{16 b}+\frac {\ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{16 b}\) | \(91\) |
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. 834 vs.
\(2 (37) = 74\).
time = 0.53, size = 834, normalized size = 19.40 \begin {gather*} \frac {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 )} \cos \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 )} \cos \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 )} \cos \left (4 \, b x + 4 \, 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 (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) - 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 )} \sin \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 )} \sin \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 )} \sin \left (4 \, b x + 4 \, a\right ) - 40 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (3 \, b x + 3 \, a\right ) + 120 \, \cos \left (3 \, b x + 3 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 36 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 36 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) - 12 \, \sin \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (37) = 74\).
time = 2.16, size = 94, normalized size = 2.19 \begin {gather*} \frac {3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{96 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \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 [A]
time = 0.47, size = 52, normalized size = 1.21 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (-\sin \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 = 0.06, size = 38, normalized size = 0.88 \begin {gather*} \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}-\frac {\frac {{\sin \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,{\sin \left (a+b\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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