Integrand size = 7, antiderivative size = 77 \[ \int \sin (\coth (a+b x)) \, dx=-\frac {\operatorname {CosIntegral}(1-\coth (a+b x)) \sin (1)}{2 b}-\frac {\operatorname {CosIntegral}(1+\coth (a+b x)) \sin (1)}{2 b}+\frac {\cos (1) \text {Si}(1-\coth (a+b x))}{2 b}+\frac {\cos (1) \text {Si}(1+\coth (a+b x))}{2 b} \]
-1/2*cos(1)*Si(-1+coth(b*x+a))/b+1/2*cos(1)*Si(1+coth(b*x+a))/b-1/2*Ci(1-c oth(b*x+a))*sin(1)/b-1/2*Ci(1+coth(b*x+a))*sin(1)/b
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77 \[ \int \sin (\coth (a+b x)) \, dx=-\frac {\operatorname {CosIntegral}(1-\coth (a+b x)) \sin (1)+\operatorname {CosIntegral}(1+\coth (a+b x)) \sin (1)-\cos (1) (\text {Si}(1-\coth (a+b x))+\text {Si}(1+\coth (a+b x)))}{2 b} \]
-1/2*(CosIntegral[1 - Coth[a + b*x]]*Sin[1] + CosIntegral[1 + Coth[a + b*x ]]*Sin[1] - Cos[1]*(SinIntegral[1 - Coth[a + b*x]] + SinIntegral[1 + Coth[ a + b*x]]))/b
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4852, 3814, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin (\coth (a+b x)) \, dx\) |
\(\Big \downarrow \) 4852 |
\(\displaystyle \frac {\int \frac {\sin (\coth (a+b x))}{1-\coth ^2(a+b x)}d\coth (a+b x)}{b}\) |
\(\Big \downarrow \) 3814 |
\(\displaystyle \frac {\int \left (\frac {\sin (\coth (a+b x))}{2 (1-\coth (a+b x))}+\frac {\sin (\coth (a+b x))}{2 (\coth (a+b x)+1)}\right )d\coth (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{2} \sin (1) \operatorname {CosIntegral}(1-\coth (a+b x))-\frac {1}{2} \sin (1) \operatorname {CosIntegral}(\coth (a+b x)+1)+\frac {1}{2} \cos (1) \text {Si}(1-\coth (a+b x))+\frac {1}{2} \cos (1) \text {Si}(\coth (a+b x)+1)}{b}\) |
(-1/2*(CosIntegral[1 - Coth[a + b*x]]*Sin[1]) - (CosIntegral[1 + Coth[a + b*x]]*Sin[1])/2 + (Cos[1]*SinIntegral[1 - Coth[a + b*x]])/2 + (Cos[1]*SinI ntegral[1 + Coth[a + b*x]])/2)/b
3.3.19.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int [ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Cot[v], x]}, -d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2* x^2), Cot[v]/d, u, x], x], x, Cot[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[ NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]
Time = 0.78 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{2}+\frac {\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{2}}{b}\) | \(58\) |
default | \(\frac {-\frac {\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{2}+\frac {\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{2}-\frac {\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{2}}{b}\) | \(58\) |
risch | \(-\frac {i {\mathrm e}^{i} \operatorname {Ei}_{1}\left (-\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{4 b}+\frac {i {\mathrm e}^{-i} \operatorname {Ei}_{1}\left (-\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}-2 i\right )}{4 b}-\frac {\operatorname {csgn}\left (\frac {{\mathrm e}^{-a}}{-{\mathrm e}^{2 b x +a}+{\mathrm e}^{-a}}\right ) \pi \,{\mathrm e}^{-i}}{4 b}-\frac {\operatorname {Si}\left (\frac {2 \,{\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{-i}}{2 b}+\frac {i \operatorname {Ei}_{1}\left (-\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{-i}}{4 b}-\frac {i {\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}+2 i\right )}{4 b}\) | \(206\) |
1/b*(-1/2*Si(coth(b*x+a)-1)*cos(1)-1/2*Ci(coth(b*x+a)-1)*sin(1)+1/2*Si(cot h(b*x+a)+1)*cos(1)-1/2*Ci(coth(b*x+a)+1)*sin(1))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.81 \[ \int \sin (\coth (a+b x)) \, dx=\frac {{\left (i \, \cos \left (1\right )^{2} - 2 \, \cos \left (1\right ) \sin \left (1\right ) - i \, \sin \left (1\right )^{2} - i\right )} \operatorname {Ci}\left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) + {\left (i \, \cos \left (1\right )^{2} - 2 \, \cos \left (1\right ) \sin \left (1\right ) - i \, \sin \left (1\right )^{2} - i\right )} \operatorname {Ci}\left (\frac {2}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right ) + {\left (\cos \left (1\right )^{2} + 2 i \, \cos \left (1\right ) \sin \left (1\right ) - \sin \left (1\right )^{2} + 1\right )} \operatorname {Si}\left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) - {\left (\cos \left (1\right )^{2} + 2 i \, \cos \left (1\right ) \sin \left (1\right ) - \sin \left (1\right )^{2} + 1\right )} \operatorname {Si}\left (\frac {2}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )}{4 \, {\left (b \cos \left (1\right ) + i \, b \sin \left (1\right )\right )}} \]
1/4*((I*cos(1)^2 - 2*cos(1)*sin(1) - I*sin(1)^2 - I)*cos_integral((cosh(b* x + a) + sinh(b*x + a))/sinh(b*x + a)) + (I*cos(1)^2 - 2*cos(1)*sin(1) - I *sin(1)^2 - I)*cos_integral(2/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)) + (cos(1)^2 + 2*I*cos(1)*sin(1) - sin(1)^2 + 1)*sin_integral((cosh(b*x + a) + sinh(b*x + a))/sinh(b*x + a)) - (cos(1)^2 + 2*I*cos(1)*sin(1) - sin(1)^2 + 1)*sin_integral(2/(cosh(b*x + a)^2 + 2*c osh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)))/(b*cos(1) + I*b*sin(1) )
\[ \int \sin (\coth (a+b x)) \, dx=\int \sin {\left (\coth {\left (a + b x \right )} \right )}\, dx \]
\[ \int \sin (\coth (a+b x)) \, dx=\int { \sin \left (\coth \left (b x + a\right )\right ) \,d x } \]
\[ \int \sin (\coth (a+b x)) \, dx=\int { \sin \left (\coth \left (b x + a\right )\right ) \,d x } \]
Timed out. \[ \int \sin (\coth (a+b x)) \, dx=\int \sin \left (\mathrm {coth}\left (a+b\,x\right )\right ) \,d x \]