Integrand size = 13, antiderivative size = 29 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=-\frac {\text {arctanh}(\cosh (c+b x)) \sinh (a-c)}{b}+\frac {\sinh (a+b x)}{b} \] Output:
-arctanh(cosh(b*x+c))*sinh(a-c)/b+sinh(b*x+a)/b
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=\frac {\cosh (b x) \sinh (a)}{b}-\frac {2 i \arctan \left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh (c) \cosh \left (\frac {b x}{2}\right )+\sinh (c) \sinh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \sinh (a-c)}{b}+\frac {\cosh (a) \sinh (b x)}{b} \] Input:
Integrate[Coth[c + b*x]*Sinh[a + b*x],x]
Output:
(Cosh[b*x]*Sinh[a])/b - ((2*I)*ArcTan[((Cosh[c] - Sinh[c])*(Cosh[c]*Cosh[( b*x)/2] + Sinh[c]*Sinh[(b*x)/2]))/(I*Cosh[c]*Cosh[(b*x)/2] - I*Cosh[(b*x)/ 2]*Sinh[c])]*Sinh[a - c])/b + (Cosh[a]*Sinh[b*x])/b
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6156, 3042, 26, 3117, 4257}
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.
\(\displaystyle \int \sinh (a+b x) \coth (b x+c) \, dx\) |
\(\Big \downarrow \) 6156 |
\(\displaystyle \sinh (a-c) \int \text {csch}(c+b x)dx+\int \cosh (a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (a-c) \int i \csc (i c+i b x)dx+\int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \sinh (a-c) \int \csc (i c+i b x)dx+\int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {\sinh (a+b x)}{b}+i \sinh (a-c) \int \csc (i c+i b x)dx\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\sinh (a+b x)}{b}-\frac {\sinh (a-c) \text {arctanh}(\cosh (b x+c))}{b}\) |
Input:
Int[Coth[c + b*x]*Sinh[a + b*x],x]
Output:
-((ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b) + Sinh[a + b*x]/b
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Coth[w_]^(n_.)*Sinh[v_], x_Symbol] :> Int[Cosh[v]*Coth[w]^(n - 1), x] + Simp[Sinh[v - w] Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ [w, v] && FreeQ[v - w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(29)=58\).
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.34
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(155\) |
Input:
int(coth(b*x+c)*sinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/2/b*exp(b*x+a)-1/2/b*exp(-b*x-a)+1/2*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c) *exp(2*a)-1/2*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*c)-1/2*ln(exp(b*x+ a)+exp(a-c))/b*exp(-a-c)*exp(2*a)+1/2*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)* exp(2*c)
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 439, normalized size of antiderivative = 15.14 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=\frac {\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} + {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) + 1\right ) - {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) - 1\right ) + 2 \, {\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right ) - 1}{2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )\right )}} \] Input:
integrate(coth(b*x+c)*sinh(b*x+a),x, algorithm="fricas")
Output:
1/2*(cosh(b*x + c)^2*cosh(-a + c)^2 - 2*cosh(b*x + c)^2*cosh(-a + c)*sinh( -a + c) + cosh(b*x + c)^2*sinh(-a + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c )*sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^2 + (2*cosh(b*x + c)*cosh(- a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)* cosh(b*x + c) - (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) - (2*cosh( b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh( -a + c)^2 - 1)*cosh(b*x + c) - (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))*log(cosh(b*x + c) + sinh(b*x + c) - 1) + 2*(cosh(b*x + c)*cosh(-a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh (-a + c) + cosh(b*x + c)*sinh(-a + c)^2)*sinh(b*x + c) - 1)/(b*cosh(b*x + c)*cosh(-a + c) - b*cosh(b*x + c)*sinh(-a + c) + (b*cosh(-a + c) - b*sinh( -a + c))*sinh(b*x + c))
\[ \int \coth (c+b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \coth {\left (b x + c \right )}\, dx \] Input:
integrate(coth(b*x+c)*sinh(b*x+a),x)
Output:
Integral(sinh(a + b*x)*coth(b*x + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (29) = 58\).
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.24 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} \] Input:
integrate(coth(b*x+c)*sinh(b*x+a),x, algorithm="maxima")
Output:
-1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) - e^c)/b + 1/2*e^(b*x + a)/b - 1/2*e^(-b *x - a)/b
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + a + c\right )} + e^{a}\right ) - {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + a + c\right )} - e^{a} \right |}\right ) - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}}{2 \, b} \] Input:
integrate(coth(b*x+c)*sinh(b*x+a),x, algorithm="giac")
Output:
-1/2*((e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + a + c) + e^a) - (e ^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs(e^(b*x + a + c) - e^a)) - e^(b* x + a) + e^(-b*x - a))/b
Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.79 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {-b^2}} \] Input:
int(coth(c + b*x)*sinh(a + b*x),x)
Output:
exp(a + b*x)/(2*b) - exp(- a - b*x)/(2*b) + (atan((exp(-a)*exp(2*c)*exp(b* x)*((-b^2)^(1/2) - exp(2*a)*exp(-2*c)*(-b^2)^(1/2)))/(b*(exp(-2*a)*exp(2*c )*(exp(4*a)*exp(-4*c) - 2*exp(2*a)*exp(-2*c) + 1))^(1/2)))*(exp(2*c - 2*a) *(exp(4*a - 4*c) - 2*exp(2*a - 2*c) + 1))^(1/2))/(-b^2)^(1/2)
Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.93 \[ \int \coth (c+b x) \sinh (a+b x) \, dx=\frac {e^{2 b x +2 a +c}+e^{b x +2 a} \mathrm {log}\left (e^{b x +c}-1\right )-e^{b x +2 a} \mathrm {log}\left (e^{b x +c}+1\right )-e^{b x +2 c} \mathrm {log}\left (e^{b x +c}-1\right )+e^{b x +2 c} \mathrm {log}\left (e^{b x +c}+1\right )-e^{c}}{2 e^{b x +a +c} b} \] Input:
int(coth(b*x+c)*sinh(b*x+a),x)
Output:
(e**(2*a + 2*b*x + c) + e**(2*a + b*x)*log(e**(b*x + c) - 1) - e**(2*a + b *x)*log(e**(b*x + c) + 1) - e**(b*x + 2*c)*log(e**(b*x + c) - 1) + e**(b*x + 2*c)*log(e**(b*x + c) + 1) - e**c)/(2*e**(a + b*x + c)*b)