Integrand size = 7, antiderivative size = 77 \[ \int \cos (\coth (a+b x)) \, dx=-\frac {\cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \operatorname {CosIntegral}(1+\coth (a+b x))}{2 b}-\frac {\sin (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\sin (1) \text {Si}(1+\coth (a+b x))}{2 b} \]
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Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3415, 3384, 3380, 3383} \[ \int \cos (\coth (a+b x)) \, dx=-\frac {\cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \operatorname {CosIntegral}(\coth (a+b x)+1)}{2 b}-\frac {\sin (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\sin (1) \text {Si}(\coth (a+b x)+1)}{2 b} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3415
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\cos (x)}{2 (1-x)}+\frac {\cos (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cos (x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b} \\ & = \frac {\cos (1) \text {Subst}\left (\int \frac {\cos (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\cos (1) \text {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\sin (1) \text {Subst}\left (\int \frac {\sin (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\sin (1) \text {Subst}\left (\int \frac {\sin (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b} \\ & = -\frac {\cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \operatorname {CosIntegral}(1+\coth (a+b x))}{2 b}-\frac {\sin (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\sin (1) \text {Si}(1+\coth (a+b x))}{2 b} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \cos (\coth (a+b x)) \, dx=-\frac {\cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))-\cos (1) \operatorname {CosIntegral}(1+\coth (a+b x))+\sin (1) \text {Si}(1-\coth (a+b x))-\sin (1) \text {Si}(1+\coth (a+b x))}{2 b} \]
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Time = 0.75 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{2}+\frac {\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{2}+\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{2}}{b}\) | \(58\) |
default | \(\frac {\frac {\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{2}+\frac {\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{2}+\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{2}}{b}\) | \(58\) |
risch | \(\frac {{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{4 b}-\frac {{\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}+2 i\right )}{4 b}-\frac {i \operatorname {csgn}\left (\frac {{\mathrm e}^{-a}}{-{\mathrm e}^{2 b x +a}+{\mathrm e}^{-a}}\right ) {\mathrm e}^{i} \pi }{4 b}-\frac {i \operatorname {Si}\left (\frac {2 \,{\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{i}}{2 b}+\frac {{\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{4 b}-\frac {{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (-\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}-2 i\right )}{4 b}\) | \(204\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.81 \[ \int \cos (\coth (a+b x)) \, dx=\frac {{\left (\cos \left (1\right )^{2} + 2 i \, \cos \left (1\right ) \sin \left (1\right ) - \sin \left (1\right )^{2} + 1\right )} \operatorname {Ci}\left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) - {\left (\cos \left (1\right )^{2} + 2 i \, \cos \left (1\right ) \sin \left (1\right ) - \sin \left (1\right )^{2} + 1\right )} \operatorname {Ci}\left (\frac {2}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right ) + {\left (-i \, \cos \left (1\right )^{2} + 2 \, \cos \left (1\right ) \sin \left (1\right ) + i \, \sin \left (1\right )^{2} + i\right )} \operatorname {Si}\left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) + {\left (-i \, \cos \left (1\right )^{2} + 2 \, \cos \left (1\right ) \sin \left (1\right ) + i \, \sin \left (1\right )^{2} + i\right )} \operatorname {Si}\left (\frac {2}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )}{4 \, {\left (b \cos \left (1\right ) + i \, b \sin \left (1\right )\right )}} \]
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\[ \int \cos (\coth (a+b x)) \, dx=\int \cos {\left (\coth {\left (a + b x \right )} \right )}\, dx \]
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\[ \int \cos (\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right ) \,d x } \]
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\[ \int \cos (\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right ) \,d x } \]
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Timed out. \[ \int \cos (\coth (a+b x)) \, dx=\int \cos \left (\mathrm {coth}\left (a+b\,x\right )\right ) \,d x \]
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