Optimal. Leaf size=77 \[ -\frac {\sin (1) \text {Ci}(1-\coth (a+b x))}{2 b}-\frac {\sin (1) \text {Ci}(\coth (a+b x)+1)}{2 b}+\frac {\cos (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \text {Si}(\coth (a+b x)+1)}{2 b} \]
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Rubi [A] time = 0.14, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 = {3333, 3303, 3299, 3302} \[ -\frac {\sin (1) \text {CosIntegral}(1-\coth (a+b x))}{2 b}-\frac {\sin (1) \text {CosIntegral}(\coth (a+b x)+1)}{2 b}+\frac {\cos (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \text {Si}(\coth (a+b x)+1)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rubi steps
\begin {align*} \int \sin (\coth (a+b x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (x)}{2 (1-x)}+\frac {\sin (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac {\cos (1) \operatorname {Subst}\left (\int \frac {\sin (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\cos (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 {\cos (1-x)}{1-x} \, dx,x,\coth (a+b x)\right )}{2 b}-\frac {\sin (1) \operatorname {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac {\text {Ci}(1-\coth (a+b x)) \sin (1)}{2 b}-\frac {\text {Ci}(1+\coth (a+b x)) \sin (1)}{2 b}+\frac {\cos (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \text {Si}(1+\coth (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 59, normalized size = 0.77 \[ -\frac {\sin (1) \text {Ci}(1-\coth (a+b x))+\sin (1) \text {Ci}(\coth (a+b x)+1)-\cos (1) (\text {Si}(1-\coth (a+b x))+\text {Si}(\coth (a+b x)+1))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 149, normalized size = 1.94 \[ -\frac {\operatorname {Ci}\left (\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) \sin \relax (1) + \operatorname {Ci}\left (-\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) \sin \relax (1) + \operatorname {Ci}\left (\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) \sin \relax (1) + \operatorname {Ci}\left (-\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) \sin \relax (1) - 2 \, \cos \relax (1) \operatorname {Si}\left (\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 2 \, \cos \relax (1) \operatorname {Si}\left (\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (\coth \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 58, normalized size = 0.75 \[ \frac {\frac {\Si \left (1+\coth \left (b x +a \right )\right ) \cos \relax (1)}{2}-\frac {\Ci \left (1+\coth \left (b x +a \right )\right ) \sin \relax (1)}{2}-\frac {\Si \left (-1+\coth \left (b x +a \right )\right ) \cos \relax (1)}{2}-\frac {\Ci \left (-1+\coth \left (b x +a \right )\right ) \sin \relax (1)}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (\coth \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sin \left (\mathrm {coth}\left (a+b\,x\right )\right ) \,d x \]
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 \[ \int \sin {\left (\coth {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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