Optimal. Leaf size=46 \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4577, 4578, 2637, 3770, 2606, 8} \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2637
Rule 3770
Rule 4577
Rule 4578
Rubi steps
\begin {align*} \int \cos (a+b x) \cot ^2(c+b x) \, dx &=\cos (a-c) \int \cot (c+b x) \csc (c+b x) \, dx-\int \cot (c+b x) \sin (a+b x) \, dx\\ &=-\frac {\cos (a-c) \operatorname {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}-\sin (a-c) \int \csc (c+b x) \, dx-\int \cos (a+b x) \, dx\\ &=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {\tanh ^{-1}(\cos (c+b x)) \sin (a-c)}{b}-\frac {\sin (a+b x)}{b}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 112, normalized size = 2.43 \[ -\frac {\cos (a-c) \csc (b x+c)}{b}+\frac {2 i \sin (a-c) \tan ^{-1}\left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{\sin (c) \cos \left (\frac {b x}{2}\right )+i \cos (c) \cos \left (\frac {b x}{2}\right )}\right )}{b}-\frac {\sin (a) \cos (b x)}{b}-\frac {\cos (a) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 316, normalized size = 6.87 \[ \frac {4 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 8 \, \cos \left (-2 \, a + 2 \, c\right ) - 8}{4 \, {\left (b \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \sin \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 145, normalized size = 3.15 \[ \frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}-\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}+\frac {i \left ({\mathrm e}^{i \left (b x +3 a \right )}+{\mathrm e}^{i \left (b x +a +2 c \right )}\right )}{b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 613, normalized size = 13.33 \[ \frac {{\left (\sin \left (3 \, b x + a + 2 \, c\right ) - \sin \left (b x + a\right )\right )} \cos \left (4 \, b x + 2 \, a + 2 \, c\right ) + 3 \, {\left (\sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, c\right )\right )} \cos \left (3 \, b x + a + 2 \, c\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) + {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right ) - \cos \left (b x + a\right )\right )} \sin \left (4 \, b x + 2 \, a + 2 \, c\right ) - {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) - 1\right )} \sin \left (3 \, b x + a + 2 \, c\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, c\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) \sin \left (b x + a\right ) - \sin \left (b x + a\right )}{2 \, {\left (b \cos \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) + b \cos \left (b x + a\right )^{2} + b \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + b \sin \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.44, size = 289, normalized size = 6.28 \[ -\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (a + b x \right )} \cot ^{2}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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