Optimal. Leaf size=36 \[ -\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4678, 2686, 8,
3855} \begin {gather*} -\frac {\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a-c) \csc (b x+c)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2686
Rule 3855
Rule 4678
Rubi steps
\begin {align*} \int \csc ^2(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc (c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc (c+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\sin (a-c) \text {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}\\ &=-\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.12, size = 90, normalized size = 2.50 \begin {gather*} -\frac {2 i \text {ArcTan}\left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{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. \(346\) vs.
\(2(36)=72\).
time = 0.62, size = 347, normalized size = 9.64
method | result | size |
risch | \(\frac {{\mathrm e}^{i \left (b x +3 a \right )}-{\mathrm e}^{i \left (b x +a +2 c \right )}}{b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\) | \(115\) |
default | \(\frac {\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )-2 \sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+8 \sin \left (a \right ) \cos \left (c \right )-8 \cos \left (a \right ) \sin \left (c \right )}{\left (-4 \left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-4 \left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )\right ) \left (\cos \left (c \right ) \sin \left (a \right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\sin \left (c \right ) \cos \left (a \right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \cos \left (a \right ) \cos \left (c \right )+2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}+\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )-2 \sin \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}}\right )}{\left (-4 \left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-4 \left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )\right ) \sqrt {-\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}}}{b}\) | \(347\) |
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. 454 vs.
\(2 (36) = 72\).
time = 0.31, size = 454, normalized size = 12.61 \begin {gather*} -\frac {2 \, {\left (\cos \left (b x + 2 \, a\right ) - \cos \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - 2 \, \cos \left (b x + 2 \, a\right ) \cos \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \cos \left (a\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \cos \left (-a + c\right )\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 ) - {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \cos \left (-a + c\right )\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 \, {\left (\sin \left (b x + 2 \, a\right ) - \sin \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) - 2 \, \sin \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \sin \left (b x + 2 \, c\right ) \sin \left (a\right )}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.73, size = 71, normalized size = 1.97 \begin {gather*} -\frac {\cos \left (-a + c\right ) \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - \cos \left (-a + c\right ) \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 2 \, \sin \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )} \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. 1690 vs.
\(2 (29) = 58\).
time = 60.43, size = 3264, normalized size = 90.67 \begin {gather*} \text {Too large to display} \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. 349 vs.
\(2 (36) = 72\).
time = 0.40, size = 349, normalized size = 9.69 \begin {gather*} \frac {\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 ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \right |}\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 {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \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} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\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 {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}{{\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} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.21, size = 252, normalized size = 7.00 \begin {gather*} -\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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