Integrand size = 27, antiderivative size = 57 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\log (\sinh (c+d x))}{a d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a b^2 d}+\frac {\sinh (c+d x)}{b d} \]
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {\log (\sinh (c+d x))}{a}-\left (\frac {1}{a}+\frac {a}{b^2}\right ) \log (a+b \sinh (c+d x))+\frac {\sinh (c+d x)}{b}}{d} \]
Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 26, 3316, 26, 27, 522, 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.
\(\displaystyle \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos (i c+i d x)^3}{\sin (i c+i d x) (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos (i c+i d x)^3}{\sin (i c+i d x) (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle -\frac {i \int \frac {i \text {csch}(c+d x) \left (\sinh ^2(c+d x) b^2+b^2\right )}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\int \frac {\text {csch}(c+d x) \left (\sinh ^2(c+d x) b^2+b^2\right )}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\text {csch}(c+d x) \left (\sinh ^2(c+d x) b^2+b^2\right )}{b (a+b \sinh (c+d x))}d(b \sinh (c+d x))}{b^2 d}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {\int \left (\frac {-a^2-b^2}{a (a+b \sinh (c+d x))}+\frac {b \text {csch}(c+d x)}{a}+1\right )d(b \sinh (c+d x))}{b^2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a}+\frac {b^2 \log (b \sinh (c+d x))}{a}+b \sinh (c+d x)}{b^2 d}\) |
((b^2*Log[b*Sinh[c + d*x]])/a - ((a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/a + b*Sinh[c + d*x])/(b^2*d)
3.5.33.3.1 Defintions of rubi rules used
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(57)=114\).
Time = 2.34 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.33
method | result | size |
risch | \(\frac {a x}{b^{2}}+\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}+\frac {2 a c}{b^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d a}\) | \(133\) |
derivativedivides | \(\frac {-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}+\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \,b^{2}}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(138\) |
default | \(\frac {-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}+\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \,b^{2}}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(138\) |
a*x/b^2+1/2/b/d*exp(d*x+c)-1/2/b/d*exp(-d*x-c)+2*a/b^2/d*c+1/d/a*ln(exp(2* d*x+2*c)-1)-a/b^2/d*ln(exp(2*d*x+2*c)+2*a/b*exp(d*x+c)-1)-1/d/a*ln(exp(2*d *x+2*c)+2*a/b*exp(d*x+c)-1)
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.56 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a^{2} d x \cosh \left (d x + c\right ) + a b \cosh \left (d x + c\right )^{2} + a b \sinh \left (d x + c\right )^{2} - a b - 2 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a^{2} d x + a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}} \]
1/2*(2*a^2*d*x*cosh(d*x + c) + a*b*cosh(d*x + c)^2 + a*b*sinh(d*x + c)^2 - a*b - 2*((a^2 + b^2)*cosh(d*x + c) + (a^2 + b^2)*sinh(d*x + c))*log(2*(b* sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(b^2*cosh(d*x + c) + b^2*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(a^2*d*x + a*b*cosh(d*x + c))*sinh(d*x + c))/(a*b^2*d*cosh(d*x + c) + a*b^2*d*sinh(d*x + c))
\[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\cosh ^{2}{\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (57) = 114\).
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.28 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (d x + c\right )} a}{b^{2} d} + \frac {e^{\left (d x + c\right )}}{2 \, b d} - \frac {e^{\left (-d x - c\right )}}{2 \, b d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a b^{2} d} \]
-(d*x + c)*a/(b^2*d) + 1/2*e^(d*x + c)/(b*d) - 1/2*e^(-d*x - c)/(b*d) + lo g(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d) - (a^2 + b^2)*log( -2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a*b^2*d)
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.65 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{b} + \frac {2 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a} - \frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a b^{2}}}{2 \, d} \]
1/2*((e^(d*x + c) - e^(-d*x - c))/b + 2*log(abs(e^(d*x + c) - e^(-d*x - c) ))/a - 2*(a^2 + b^2)*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a*b^2 ))/d
Time = 1.51 (sec) , antiderivative size = 360, normalized size of antiderivative = 6.32 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d}-\frac {\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}+\frac {\ln \left (4\,a^6+16\,b^6+32\,a^2\,b^4+20\,a^4\,b^2-4\,a^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-16\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-20\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a\,d}+\frac {a\,x}{b^2}-\frac {a\,\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{b^2\,d} \]
exp(c + d*x)/(2*b*d) - exp(- c - d*x)/(2*b*d) - log(8*a^5*exp(d*x)*exp(c) - 16*b^5 - 16*a^2*b^3 - 4*a^4*b + 16*b^5*exp(2*c)*exp(2*d*x) + 4*a^4*b*exp (2*c)*exp(2*d*x) + 32*a^3*b^2*exp(d*x)*exp(c) + 16*a^2*b^3*exp(2*c)*exp(2* d*x) + 32*a*b^4*exp(d*x)*exp(c))/(a*d) + log(4*a^6 + 16*b^6 + 32*a^2*b^4 + 20*a^4*b^2 - 4*a^6*exp(2*c)*exp(2*d*x) - 16*b^6*exp(2*c)*exp(2*d*x) - 32* a^2*b^4*exp(2*c)*exp(2*d*x) - 20*a^4*b^2*exp(2*c)*exp(2*d*x))/(a*d) + (a*x )/b^2 - (a*log(8*a^5*exp(d*x)*exp(c) - 16*b^5 - 16*a^2*b^3 - 4*a^4*b + 16* b^5*exp(2*c)*exp(2*d*x) + 4*a^4*b*exp(2*c)*exp(2*d*x) + 32*a^3*b^2*exp(d*x )*exp(c) + 16*a^2*b^3*exp(2*c)*exp(2*d*x) + 32*a*b^4*exp(d*x)*exp(c)))/(b^ 2*d)