Optimal. Leaf size=77 \[ -\frac{\cos (1) \text{CosIntegral}(1-\coth (a+b x))}{2 b}+\frac{\cos (1) \text{CosIntegral}(\coth (a+b x)+1)}{2 b}-\frac{\sin (1) \text{Si}(1-\coth (a+b x))}{2 b}+\frac{\sin (1) \text{Si}(\coth (a+b x)+1)}{2 b} \]
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Rubi [A] time = 0.140022, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {3334, 3303, 3299, 3302} \[ -\frac{\cos (1) \text{CosIntegral}(1-\coth (a+b x))}{2 b}+\frac{\cos (1) \text{CosIntegral}(\coth (a+b x)+1)}{2 b}-\frac{\sin (1) \text{Si}(1-\coth (a+b x))}{2 b}+\frac{\sin (1) \text{Si}(\coth (a+b x)+1)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3334
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \cos (\coth (a+b x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac{\operatorname{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{\operatorname{Subst}\left (\int \frac{\cos (x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=\frac{\cos (1) \operatorname{Subst}\left (\int \frac{\cos (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac{\cos (1) \operatorname{Subst}\left (\int \frac{\cos (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac{\sin (1) \operatorname{Subst}\left (\int \frac{\sin (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac{\sin (1) \operatorname{Subst}\left (\int \frac{\sin (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac{\cos (1) \text{Ci}(1-\coth (a+b x))}{2 b}+\frac{\cos (1) \text{Ci}(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*}
Mathematica [A] time = 0.105566, size = 62, normalized size = 0.81 \[ -\frac{\cos (1) \text{CosIntegral}(1-\coth (a+b x))-\cos (1) \text{CosIntegral}(\coth (a+b x)+1)+\sin (1) \text{Si}(1-\coth (a+b x))-\sin (1) \text{Si}(\coth (a+b x)+1)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 58, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Si} \left ({\rm coth} \left (bx+a\right )+1 \right ) \sin \left ( 1 \right ) }{2}}+{\frac{{\it Ci} \left ({\rm coth} \left (bx+a\right )+1 \right ) \cos \left ( 1 \right ) }{2}}+{\frac{{\it Si} \left ({\rm coth} \left (bx+a\right )-1 \right ) \sin \left ( 1 \right ) }{2}}-{\frac{{\it Ci} \left ({\rm coth} \left (bx+a\right )-1 \right ) \cos \left ( 1 \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (\coth \left (b x + a\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.69787, size = 463, normalized size = 6.01 \begin{align*} \frac{\cos \left (1\right ) \operatorname{Ci}\left (\frac{2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + \cos \left (1\right ) \operatorname{Ci}\left (-\frac{2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - \cos \left (1\right ) \operatorname{Ci}\left (\frac{2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - \cos \left (1\right ) \operatorname{Ci}\left (-\frac{2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 2 \, \sin \left (1\right ) \operatorname{Si}\left (\frac{2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 2 \, \sin \left (1\right ) \operatorname{Si}\left (\frac{2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (\coth{\left (a + b x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (\coth \left (b x + a\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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