Integrand size = 9, antiderivative size = 157 \[ \int \cos ^3(\coth (a+b x)) \, dx=-\frac {\cos (3) \operatorname {CosIntegral}(3-3 \coth (a+b x))}{8 b}-\frac {3 \cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))}{8 b}+\frac {3 \cos (1) \operatorname {CosIntegral}(1+\coth (a+b x))}{8 b}+\frac {\cos (3) \operatorname {CosIntegral}(3+3 \coth (a+b x))}{8 b}-\frac {\sin (3) \text {Si}(3-3 \coth (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\coth (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(1+\coth (a+b x))}{8 b}+\frac {\sin (3) \text {Si}(3+3 \coth (a+b x))}{8 b} \]
-3/8*Ci(1-coth(b*x+a))*cos(1)/b+3/8*Ci(1+coth(b*x+a))*cos(1)/b-1/8*Ci(3-3* coth(b*x+a))*cos(3)/b+1/8*Ci(3+3*coth(b*x+a))*cos(3)/b+3/8*Si(-1+coth(b*x+ a))*sin(1)/b+3/8*Si(1+coth(b*x+a))*sin(1)/b+1/8*Si(-3+3*coth(b*x+a))*sin(3 )/b+1/8*Si(3+3*coth(b*x+a))*sin(3)/b
Time = 0.68 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int \cos ^3(\coth (a+b x)) \, dx=\frac {-2 \cos (3) \operatorname {CosIntegral}(3-3 \coth (a+b x))-6 \cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))+6 \cos (1) \operatorname {CosIntegral}(1+\coth (a+b x))+2 \cos (3) \operatorname {CosIntegral}(3+3 \coth (a+b x))-2 \sin (3) \text {Si}(3-3 \coth (a+b x))-6 \sin (1) \text {Si}(1-\coth (a+b x))+6 \sin (1) \text {Si}(1+\coth (a+b x))+2 \sin (3) \text {Si}(3+3 \coth (a+b x))}{16 b} \]
(-2*Cos[3]*CosIntegral[3 - 3*Coth[a + b*x]] - 6*Cos[1]*CosIntegral[1 - Cot h[a + b*x]] + 6*Cos[1]*CosIntegral[1 + Coth[a + b*x]] + 2*Cos[3]*CosIntegr al[3 + 3*Coth[a + b*x]] - 2*Sin[3]*SinIntegral[3 - 3*Coth[a + b*x]] - 6*Si n[1]*SinIntegral[1 - Coth[a + b*x]] + 6*Sin[1]*SinIntegral[1 + Coth[a + b* x]] + 2*Sin[3]*SinIntegral[3 + 3*Coth[a + b*x]])/(16*b)
Time = 0.58 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4852, 7276, 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.
\(\displaystyle \int \cos ^3(\coth (a+b x)) \, dx\) |
\(\Big \downarrow \) 4852 |
\(\displaystyle \frac {\int \frac {\cos ^3(\coth (a+b x))}{1-\coth ^2(a+b x)}d\coth (a+b x)}{b}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {\int \left (\frac {\cos ^3(\coth (a+b x))}{2 (\coth (a+b x)+1)}-\frac {\cos ^3(\coth (a+b x))}{2 (\coth (a+b x)-1)}\right )d\coth (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{8} \cos (3) \operatorname {CosIntegral}(3-3 \coth (a+b x))-\frac {3}{8} \cos (1) \operatorname {CosIntegral}(1-\coth (a+b x))+\frac {3}{8} \cos (1) \operatorname {CosIntegral}(\coth (a+b x)+1)+\frac {1}{8} \cos (3) \operatorname {CosIntegral}(3 \coth (a+b x)+3)-\frac {1}{8} \sin (3) \text {Si}(3-3 \coth (a+b x))-\frac {3}{8} \sin (1) \text {Si}(1-\coth (a+b x))+\frac {3}{8} \sin (1) \text {Si}(\coth (a+b x)+1)+\frac {1}{8} \sin (3) \text {Si}(3 \coth (a+b x)+3)}{b}\) |
(-1/8*(Cos[3]*CosIntegral[3 - 3*Coth[a + b*x]]) - (3*Cos[1]*CosIntegral[1 - Coth[a + b*x]])/8 + (3*Cos[1]*CosIntegral[1 + Coth[a + b*x]])/8 + (Cos[3 ]*CosIntegral[3 + 3*Coth[a + b*x]])/8 - (Sin[3]*SinIntegral[3 - 3*Coth[a + b*x]])/8 - (3*Sin[1]*SinIntegral[1 - Coth[a + b*x]])/8 + (3*Sin[1]*SinInt egral[1 + Coth[a + b*x]])/8 + (Sin[3]*SinIntegral[3 + 3*Coth[a + b*x]])/8) /b
3.3.21.3.1 Defintions of rubi rules used
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]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Si}\left (-3+3 \coth \left (b x +a \right )\right ) \sin \left (3\right )}{8}-\frac {\operatorname {Ci}\left (-3+3 \coth \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {\operatorname {Si}\left (3+3 \coth \left (b x +a \right )\right ) \sin \left (3\right )}{8}+\frac {\operatorname {Ci}\left (3+3 \coth \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {3 \,\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{8}-\frac {3 \,\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{8}+\frac {3 \,\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{8}+\frac {3 \,\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{8}}{b}\) | \(118\) |
default | \(\frac {\frac {\operatorname {Si}\left (-3+3 \coth \left (b x +a \right )\right ) \sin \left (3\right )}{8}-\frac {\operatorname {Ci}\left (-3+3 \coth \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {\operatorname {Si}\left (3+3 \coth \left (b x +a \right )\right ) \sin \left (3\right )}{8}+\frac {\operatorname {Ci}\left (3+3 \coth \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {3 \,\operatorname {Si}\left (\coth \left (b x +a \right )-1\right ) \sin \left (1\right )}{8}-\frac {3 \,\operatorname {Ci}\left (\coth \left (b x +a \right )-1\right ) \cos \left (1\right )}{8}+\frac {3 \,\operatorname {Si}\left (\coth \left (b x +a \right )+1\right ) \sin \left (1\right )}{8}+\frac {3 \,\operatorname {Ci}\left (\coth \left (b x +a \right )+1\right ) \cos \left (1\right )}{8}}{b}\) | \(118\) |
risch | \(\frac {{\mathrm e}^{-3 i} \operatorname {Ei}_{1}\left (\frac {6 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{16 b}-\frac {{\mathrm e}^{3 i} \operatorname {Ei}_{1}\left (\frac {6 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}+6 i\right )}{16 b}-\frac {{\mathrm e}^{-3 i} \operatorname {Ei}_{1}\left (-\frac {6 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}-6 i\right )}{16 b}-\frac {i \operatorname {csgn}\left (\frac {{\mathrm e}^{-a}}{-{\mathrm e}^{2 b x +a}+{\mathrm e}^{-a}}\right ) \pi \,{\mathrm e}^{3 i}}{16 b}-\frac {i \operatorname {Si}\left (\frac {6 \,{\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{3 i}}{8 b}+\frac {\operatorname {Ei}_{1}\left (\frac {6 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{3 i}}{16 b}-\frac {3 \,{\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}+2 i\right )}{16 b}+\frac {3 \,{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{16 b}-\frac {3 i \operatorname {csgn}\left (\frac {{\mathrm e}^{-a}}{-{\mathrm e}^{2 b x +a}+{\mathrm e}^{-a}}\right ) {\mathrm e}^{i} \pi }{16 b}-\frac {3 i \operatorname {Si}\left (\frac {2 \,{\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right ) {\mathrm e}^{i}}{8 b}+\frac {3 \,{\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}\right )}{16 b}-\frac {3 \,{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (-\frac {2 i {\mathrm e}^{-a}}{{\mathrm e}^{2 b x +a}-{\mathrm e}^{-a}}-2 i\right )}{16 b}\) | \(406\) |
1/b*(1/8*Si(-3+3*coth(b*x+a))*sin(3)-1/8*Ci(-3+3*coth(b*x+a))*cos(3)+1/8*S i(3+3*coth(b*x+a))*sin(3)+1/8*Ci(3+3*coth(b*x+a))*cos(3)+3/8*Si(coth(b*x+a )-1)*sin(1)-3/8*Ci(coth(b*x+a)-1)*cos(1)+3/8*Si(coth(b*x+a)+1)*sin(1)+3/8* Ci(coth(b*x+a)+1)*cos(1))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.45 \[ \int \cos ^3(\coth (a+b x)) \, dx=\text {Too large to display} \]
1/16*((cos(3)^2*cos(1) - (cos(1) + I*sin(1))*sin(3)^2 + 2*I*(cos(3)*cos(1) + I*cos(3)*sin(1))*sin(3) + I*(cos(3)^2 + 1)*sin(1) + cos(1))*cos_integra l(3*(cosh(b*x + a) + sinh(b*x + a))/sinh(b*x + a)) - 3*(-2*I*cos(3)*cos(1) *sin(1) + cos(3)*sin(1)^2 - (cos(1)^2 + 1)*cos(3) - I*(cos(1)^2 + 2*I*cos( 1)*sin(1) - sin(1)^2 + 1)*sin(3))*cos_integral((cosh(b*x + a) + sinh(b*x + a))/sinh(b*x + a)) - (cos(3)^2*cos(1) - (cos(1) + I*sin(1))*sin(3)^2 + 2* I*(cos(3)*cos(1) + I*cos(3)*sin(1))*sin(3) + I*(cos(3)^2 + 1)*sin(1) + cos (1))*cos_integral(6/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sin h(b*x + a)^2 - 1)) - 3*(2*I*cos(3)*cos(1)*sin(1) - cos(3)*sin(1)^2 + (cos( 1)^2 + 1)*cos(3) + I*(cos(1)^2 + 2*I*cos(1)*sin(1) - sin(1)^2 + 1)*sin(3)) *cos_integral(2/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b* x + a)^2 - 1)) + (-I*cos(3)^2*cos(1) - (-I*cos(1) + sin(1))*sin(3)^2 - 2*I *(I*cos(3)*cos(1) - cos(3)*sin(1))*sin(3) + I*(-I*cos(3)^2 + I)*sin(1) + I *cos(1))*sin_integral(3*(cosh(b*x + a) + sinh(b*x + a))/sinh(b*x + a)) + 3 *(2*cos(3)*cos(1)*sin(1) + I*cos(3)*sin(1)^2 - (I*cos(1)^2 - I)*cos(3) - I *(I*cos(1)^2 - 2*cos(1)*sin(1) - I*sin(1)^2 - I)*sin(3))*sin_integral((cos h(b*x + a) + sinh(b*x + a))/sinh(b*x + a)) + (-I*cos(3)^2*cos(1) - (-I*cos (1) + sin(1))*sin(3)^2 - 2*I*(I*cos(3)*cos(1) - cos(3)*sin(1))*sin(3) + I* (-I*cos(3)^2 + I)*sin(1) + I*cos(1))*sin_integral(6/(cosh(b*x + a)^2 + 2*c osh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)) + 3*(2*cos(3)*cos(1)...
\[ \int \cos ^3(\coth (a+b x)) \, dx=\int \cos ^{3}{\left (\coth {\left (a + b x \right )} \right )}\, dx \]
\[ \int \cos ^3(\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right )^{3} \,d x } \]
\[ \int \cos ^3(\coth (a+b x)) \, dx=\int { \cos \left (\coth \left (b x + a\right )\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(\coth (a+b x)) \, dx=\int {\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )}^3 \,d x \]