Optimal. Leaf size=27 \[ \frac {\cos (a-c) \log (\sin (c+b x))}{b}-x \sin (a-c) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4677, 3556, 8}
\begin {gather*} \frac {\cos (a-c) \log (\sin (b x+c))}{b}-x \sin (a-c) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3556
Rule 4677
Rubi steps
\begin {align*} \int \cos (a+b x) \csc (c+b x) \, dx &=\cos (a-c) \int \cot (c+b x) \, dx-\sin (a-c) \int 1 \, dx\\ &=\frac {\cos (a-c) \log (\sin (c+b x))}{b}-x \sin (a-c)\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 58, normalized size = 2.15 \begin {gather*} \frac {-2 i \text {ArcTan}(\tan (c+b x)) \cos (a-c)+\cos (a-c) \left (2 i b x+\log \left (\sin ^2(c+b x)\right )\right )-2 b x \sin (a-c)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs.
\(2(27)=54\).
time = 0.38, size = 162, normalized size = 6.00
method | result | size |
risch | \(i x \,{\mathrm e}^{i \left (a -c \right )}-2 i \cos \left (a -c \right ) x -\frac {2 i \cos \left (a -c \right ) a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\) | \(70\) |
default | \(\frac {\frac {\frac {\left (-\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right ) \ln \left (1+\tan ^{2}\left (b x +a \right )\right )}{2}+\left (-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\tan \left (b x +a \right )\right )}{\left (\cos ^{2}\left (c \right )+\sin ^{2}\left (c \right )\right ) \left (\cos ^{2}\left (a \right )+\sin ^{2}\left (a \right )\right )}+\frac {\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )}{\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )+\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )+\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )+\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}}{b}\) | \(162\) |
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. 106 vs.
\(2 (27) = 54\).
time = 0.29, size = 106, normalized size = 3.93 \begin {gather*} \frac {2 \, b x \sin \left (-a + c\right ) + \cos \left (-a + c\right ) \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + \cos \left (-a + c\right ) \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.65, size = 30, normalized size = 1.11 \begin {gather*} \frac {b x \sin \left (-a + c\right ) + \cos \left (-a + c\right ) \log \left (\frac {1}{2} \, \sin \left (b x + c\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs.
\(2 (20) = 40\).
time = 4.00, size = 335, normalized size = 12.41 \begin {gather*} - \left (\begin {cases} 0 & \text {for}\: b = 0 \wedge \left (b = 0 \vee c = 0\right ) \\x & \text {for}\: c = 0 \\- \frac {b x \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {b x}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {2 \log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \sin {\left (a \right )} + \left (\begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge c = 0 \\\frac {x}{\sin {\left (c \right )}} & \text {for}\: b = 0 \\\frac {\log {\left (\sin {\left (b x \right )} \right )}}{b} & \text {for}\: c = 0 \\\frac {2 b x \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \cos {\left (a \right )} \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. 482 vs.
\(2 (27) = 54\).
time = 0.44, size = 482, normalized size = 17.85 \begin {gather*} -\frac {\frac {4 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} {\left (b x + a\right )}}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left (\tan \left (b x + a\right )^{2} + 1\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} + 8 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{3} - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, a\right )^{4} - 8 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) + 20 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 8 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) + 2 \, \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} + 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right )^{4} + 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, c\right )^{4} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 1}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 115, normalized size = 4.26 \begin {gather*} -x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )-x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )+\frac {\ln \left (-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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