Integrand size = 15, antiderivative size = 46 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=-\frac {\text {arctanh}(\cosh (c+b x)) \cosh (a-c)}{b}+\frac {\cosh (a+b x)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b} \] Output:
-arctanh(cosh(b*x+c))*cosh(a-c)/b+cosh(b*x+a)/b-csch(b*x+c)*sinh(a-c)/b
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.39 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh (c) \cosh \left (\frac {b x}{2}\right )+\sinh (c) \sinh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \cosh (a-c)}{b}+\frac {\cosh (a) \cosh (b x)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b}+\frac {\sinh (a) \sinh (b x)}{b} \] Input:
Integrate[Coth[c + b*x]^2*Sinh[a + b*x],x]
Output:
((-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])]*Cosh[a - c ])/b + (Cosh[a]*Cosh[b*x])/b - (Csch[c + b*x]*Sinh[a - c])/b + (Sinh[a]*Si nh[b*x])/b
Time = 0.44 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6156, 3042, 3086, 24, 6155, 3042, 26, 3118, 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.
\(\displaystyle \int \sinh (a+b x) \coth ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 6156 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx+\sinh (a-c) \int \coth (c+b x) \text {csch}(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx+\sinh (a-c) \int \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx-\frac {i \sinh (a-c) \int 1d(-i \text {csch}(c+b x))}{b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 6155 |
\(\displaystyle \cosh (a-c) \int \text {csch}(c+b x)dx+\int \sinh (a+b x)dx-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cosh (a-c) \int i \csc (i c+i b x)dx+\int -i \sin (i a+i b x)dx-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \cosh (a-c) \int \csc (i c+i b x)dx-i \int \sin (i a+i b x)dx-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle i \cosh (a-c) \int \csc (i c+i b x)dx-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}+\frac {\cosh (a+b x)}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\cosh (a-c) \text {arctanh}(\cosh (b x+c))}{b}-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}+\frac {\cosh (a+b x)}{b}\) |
Input:
Int[Coth[c + b*x]^2*Sinh[a + b*x],x]
Output:
-((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b - (Csch[c + b* x]*Sinh[a - c])/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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cosh[v_]*Coth[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Coth[w]^(n - 1), x] + Simp[Cosh[v - w] Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ [w, v] && FreeQ[v - w, 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. \(196\) vs. \(2(46)=92\).
Time = 0.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 4.28
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}{b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(197\) |
Input:
int(coth(b*x+c)^2*sinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/2/b*exp(b*x+a)+1/2/b*exp(-b*x-a)+1/b*exp(b*x+a)*(exp(2*a)-exp(2*c))/(-ex p(2*b*x+2*a+2*c)+exp(2*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. 1237 vs. \(2 (46) = 92\).
Time = 0.11 (sec) , antiderivative size = 1237, normalized size of antiderivative = 26.89 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(coth(b*x+c)^2*sinh(b*x+a),x, algorithm="fricas")
Output:
1/2*(cosh(b*x + c)^4*cosh(-a + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sin h(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^4 + 4*(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)^3 - 3*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + 3*(2*cosh (b*x + c)^2*cosh(-a + c)^2 + (2*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh (-a + c)^2 - 2*(2*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c ) + 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^4 - 3*cosh(b*x + c)^2)*sinh(-a + c )^2 - ((cosh(-a + c)^2 + 1)*cosh(b*x + c)^3 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 + 1)*sinh(b*x + c)^3 - 3*(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))*sinh(b*x + c)^2 + (cosh(b*x + c)^3 - cosh(b*x + c) )*sinh(-a + c)^2 - (cosh(-a + c)^2 + 1)*cosh(b*x + c) + (3*(cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + (3*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(3*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) - 1 )*sinh(b*x + c) - 2*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) + ((cosh(-a + c )^2 + 1)*cosh(b*x + c)^3 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 + 1)*sinh(b*x + c)^3 - 3*(2*cosh(b*x + c)*cosh(-a + c)*sin h(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 + 1)*cosh(b*x + c))*sinh(b*x + c)^2 + (cosh(b*x + c)^3 - cosh(b*x + c))*sinh(-a + c)^2...
\[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \coth ^{2}{\left (b x + c \right )}\, dx \] Input:
integrate(coth(b*x+c)**2*sinh(b*x+a),x)
Output:
Integral(sinh(a + b*x)*coth(b*x + c)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (46) = 92\).
Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.04 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (3 \, e^{\left (2 \, a\right )} - 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 2 \, c\right )} - e^{\left (-3 \, b x - a\right )}\right )}} \] Input:
integrate(coth(b*x+c)^2*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*((3* e^(2*a) - 2*e^(2*c))*e^(-2*b*x - 2*a) - e^(2*c))/(b*(e^(-b*x - a + 2*c) - e^(-3*b*x - a)))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (46) = 92\).
Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.96 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + a + c\right )} + e^{a}\right ) - {\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + a + c\right )} - e^{a} \right |}\right ) + \frac {2 \, e^{\left (2 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )}}{e^{\left (3 \, b x + 3 \, a + 2 \, c\right )} - e^{\left (b x + 3 \, a\right )}} - e^{\left (b x + a\right )}}{2 \, b} \] Input:
integrate(coth(b*x+c)^2*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)) + (2*e^ (2*b*x + 4*a) - 3*e^(2*b*x + 2*a + 2*c) + e^(2*a))/(e^(3*b*x + 3*a + 2*c) - e^(b*x + 3*a)) - e^(b*x + a))/b
Time = 1.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 3.93 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left (2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left (2\,{\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{4\,a-4\,c}+1\right )}}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}-1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}-{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \] Input:
int(coth(c + b*x)^2*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 )*(2*exp(2*a)*exp(-2*c) + exp(4*a)*exp(-4*c) + 1))^(1/2)))*(exp(2*c - 2*a) *(2*exp(2*a - 2*c) + exp(4*a - 4*c) + 1))^(1/2))/(-b^2)^(1/2) + (exp(a + b *x)*(exp(2*a - 2*c) - 1))/(b*(exp(2*a - 2*c) - exp(2*a + 2*b*x)))
Time = 0.24 (sec) , antiderivative size = 247, normalized size of antiderivative = 5.37 \[ \int \coth ^2(c+b x) \sinh (a+b x) \, dx=\frac {e^{4 b x +2 a +3 c}+e^{3 b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}-1\right )-e^{3 b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}+1\right )+e^{3 b x +4 c} \mathrm {log}\left (e^{b x +c}-1\right )-e^{3 b x +4 c} \mathrm {log}\left (e^{b x +c}+1\right )-3 e^{2 b x +2 a +c}+3 e^{2 b x +3 c}-e^{b x +2 a} \mathrm {log}\left (e^{b x +c}-1\right )+e^{b x +2 a} \mathrm {log}\left (e^{b x +c}+1\right )-e^{b x +2 c} \mathrm {log}\left (e^{b x +c}-1\right )+e^{b x +2 c} \mathrm {log}\left (e^{b x +c}+1\right )-e^{c}}{2 e^{b x +a +c} b \left (e^{2 b x +2 c}-1\right )} \] Input:
int(coth(b*x+c)^2*sinh(b*x+a),x)
Output:
(e**(2*a + 4*b*x + 3*c) + e**(2*a + 3*b*x + 2*c)*log(e**(b*x + c) - 1) - e **(2*a + 3*b*x + 2*c)*log(e**(b*x + c) + 1) + e**(3*b*x + 4*c)*log(e**(b*x + c) - 1) - e**(3*b*x + 4*c)*log(e**(b*x + c) + 1) - 3*e**(2*a + 2*b*x + c) + 3*e**(2*b*x + 3*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*(e**(2*b*x + 2*c) - 1))