Integrand size = 9, antiderivative size = 115 \[ \int \cos ^2(\coth (a+b x)) \, dx=-\frac {\cos (2) \operatorname {CosIntegral}(2-2 \coth (a+b x))}{4 b}+\frac {\cos (2) \operatorname {CosIntegral}(2+2 \coth (a+b x))}{4 b}-\frac {\log (1-\coth (a+b x))}{4 b}+\frac {\log (1+\coth (a+b x))}{4 b}-\frac {\sin (2) \text {Si}(2-2 \coth (a+b x))}{4 b}+\frac {\sin (2) \text {Si}(2+2 \coth (a+b x))}{4 b} \]
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Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6857, 3393, 3384, 3380, 3383} \[ \int \cos ^2(\coth (a+b x)) \, dx=-\frac {\cos (2) \operatorname {CosIntegral}(2-2 \coth (a+b x))}{4 b}+\frac {\cos (2) \operatorname {CosIntegral}(2 \coth (a+b x)+2)}{4 b}-\frac {\sin (2) \text {Si}(2-2 \coth (a+b x))}{4 b}+\frac {\sin (2) \text {Si}(2 \coth (a+b x)+2)}{4 b}-\frac {\log (1-\coth (a+b x))}{4 b}+\frac {\log (\coth (a+b x)+1)}{4 b} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\cos ^2(x)}{2 (-1+x)}+\frac {\cos ^2(x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{2 (-1+x)}+\frac {\cos (2 x)}{2 (-1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 (1+x)}+\frac {\cos (2 x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{2 b} \\ & = -\frac {\log (1-\coth (a+b x))}{4 b}+\frac {\log (1+\coth (a+b x))}{4 b}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{4 b}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{4 b} \\ & = -\frac {\log (1-\coth (a+b x))}{4 b}+\frac {\log (1+\coth (a+b x))}{4 b}-\frac {\cos (2) \text {Subst}\left (\int \frac {\cos (2-2 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{4 b}+\frac {\cos (2) \text {Subst}\left (\int \frac {\cos (2+2 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{4 b}-\frac {\sin (2) \text {Subst}\left (\int \frac {\sin (2-2 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{4 b}+\frac {\sin (2) \text {Subst}\left (\int \frac {\sin (2+2 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{4 b} \\ & = -\frac {\cos (2) \operatorname {CosIntegral}(2-2 \coth (a+b x))}{4 b}+\frac {\cos (2) \operatorname {CosIntegral}(2+2 \coth (a+b x))}{4 b}-\frac {\log (1-\coth (a+b x))}{4 b}+\frac {\log (1+\coth (a+b x))}{4 b}-\frac {\sin (2) \text {Si}(2-2 \coth (a+b x))}{4 b}+\frac {\sin (2) \text {Si}(2+2 \coth (a+b x))}{4 b} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \cos ^2(\coth (a+b x)) \, dx=\frac {-\cos (2) \operatorname {CosIntegral}(2-2 \coth (a+b x))+\cos (2) \operatorname {CosIntegral}(2 (1+\coth (a+b x)))-\log (1-\coth (a+b x))+\log (1+\coth (a+b x))-\sin (2) \text {Si}(2-2 \coth (a+b x))+\sin (2) \text {Si}(2 (1+\coth (a+b x)))}{4 b} \]
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Time = 1.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Si}\left (2+2 \coth \left (b x +a \right )\right ) \sin \left (2\right )}{4}+\frac {\operatorname {Ci}\left (2+2 \coth \left (b x +a \right )\right ) \cos \left (2\right )}{4}+\frac {\operatorname {Si}\left (-2+2 \coth \left (b x +a \right )\right ) \sin \left (2\right )}{4}-\frac {\operatorname {Ci}\left (-2+2 \coth \left (b x +a \right )\right ) \cos \left (2\right )}{4}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{4}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{4}}{b}\) | \(88\) |
default | \(\frac {\frac {\operatorname {Si}\left (2+2 \coth \left (b x +a \right )\right ) \sin \left (2\right )}{4}+\frac {\operatorname {Ci}\left (2+2 \coth \left (b x +a \right )\right ) \cos \left (2\right )}{4}+\frac {\operatorname {Si}\left (-2+2 \coth \left (b x +a \right )\right ) \sin \left (2\right )}{4}-\frac {\operatorname {Ci}\left (-2+2 \coth \left (b x +a \right )\right ) \cos \left (2\right )}{4}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{4}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{4}}{b}\) | \(88\) |
risch | \(-\frac {{\mathrm e}^{2 i} \operatorname {Ei}_{1}\left (\frac {4 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}+4 i\right )}{8 b}+\frac {{\mathrm e}^{-2 i} \operatorname {Ei}_{1}\left (\frac {4 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{8 b}-\frac {{\mathrm e}^{-2 i} \operatorname {Ei}_{1}\left (-\frac {4 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}-4 i\right )}{8 b}-\frac {i {\mathrm e}^{2 i} \operatorname {csgn}\left (\frac {{\mathrm e}^{-a}}{-{\mathrm e}^{2 b x +a}+{\mathrm e}^{-a}}\right ) \pi }{8 b}-\frac {i {\mathrm e}^{2 i} \operatorname {Si}\left (\frac {4 \,{\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{4 b}+\frac {{\mathrm e}^{2 i} \operatorname {Ei}_{1}\left (\frac {4 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{b x}\right )}{2 b}\) | \(214\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.00 \[ \int \cos ^2(\coth (a+b x)) \, dx=\frac {4 \, b x \cos \left (2\right ) + 4 i \, b x \sin \left (2\right ) + {\left (\cos \left (2\right )^{2} + 2 i \, \cos \left (2\right ) \sin \left (2\right ) - \sin \left (2\right )^{2} + 1\right )} \operatorname {Ci}\left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\sinh \left (b x + a\right )}\right ) - {\left (\cos \left (2\right )^{2} + 2 i \, \cos \left (2\right ) \sin \left (2\right ) - \sin \left (2\right )^{2} + 1\right )} \operatorname {Ci}\left (\frac {4}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right ) + {\left (-i \, \cos \left (2\right )^{2} + 2 \, \cos \left (2\right ) \sin \left (2\right ) + i \, \sin \left (2\right )^{2} + i\right )} \operatorname {Si}\left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\sinh \left (b x + a\right )}\right ) + {\left (-i \, \cos \left (2\right )^{2} + 2 \, \cos \left (2\right ) \sin \left (2\right ) + i \, \sin \left (2\right )^{2} + i\right )} \operatorname {Si}\left (\frac {4}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )}{8 \, {\left (b \cos \left (2\right ) + i \, b \sin \left (2\right )\right )}} \]
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\[ \int \cos ^2(\coth (a+b x)) \, dx=\int \cos ^{2}{\left (\coth {\left (a + b x \right )} \right )}\, dx \]
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\[ \int \cos ^2(\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right )^{2} \,d x } \]
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\[ \int \cos ^2(\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(\coth (a+b x)) \, dx=\int {\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )}^2 \,d x \]
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