Integrand size = 27, antiderivative size = 59 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2 b d} \] Output:
-csch(d*x+c)/a/d-b*ln(sinh(d*x+c))/a^2/d+(a^2+b^2)*ln(a+b*sinh(d*x+c))/a^2 /b/d
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-a b \text {csch}(c+d x)-b^2 \log (\sinh (c+d x))+\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2 b d} \] Input:
Integrate[(Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
(-(a*b*Csch[c + d*x]) - b^2*Log[Sinh[c + d*x]] + (a^2 + b^2)*Log[a + b*Sin h[c + d*x]])/(a^2*b*d)
Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 25, 3316, 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.
\(\displaystyle \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos (i c+i d x)^3}{\sin (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos (i c+i d x)^3}{\sin (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle \frac {\int \frac {\text {csch}^2(c+d x) \left (\sinh ^2(c+d x) b^2+b^2\right )}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\text {csch}^2(c+d x) \left (\sinh ^2(c+d x) b^2+b^2\right )}{b^2 (a+b \sinh (c+d x))}d(b \sinh (c+d x))}{b d}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {\int \left (\frac {\text {csch}^2(c+d x)}{a}-\frac {b \text {csch}(c+d x)}{a^2}+\frac {a^2+b^2}{a^2 (a+b \sinh (c+d x))}\right )d(b \sinh (c+d x))}{b d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {b^2 \log (b \sinh (c+d x))}{a^2}+\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2}-\frac {b \text {csch}(c+d x)}{a}}{b d}\) |
Input:
Int[(Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
(-((b*Csch[c + d*x])/a) - (b^2*Log[b*Sinh[c + d*x]])/a^2 + ((a^2 + b^2)*Lo g[a + b*Sinh[c + d*x]])/a^2)/(b*d)
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. \(127\) vs. \(2(59)=118\).
Time = 0.81 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.17
method | result | size |
risch | \(-\frac {x}{b}-\frac {2 c}{b d}-\frac {2 \,{\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{2} d}\) | \(128\) |
derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\left (2 a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{2 b \,a^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(135\) |
default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\left (2 a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{2 b \,a^{2}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(135\) |
Input:
int(cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-x/b-2/b/d*c-2/d/a*exp(d*x+c)/(exp(2*d*x+2*c)-1)-b/a^2/d*ln(exp(2*d*x+2*c) -1)+1/b/d*ln(exp(2*d*x+2*c)+2/b*a*exp(d*x+c)-1)+b/a^2/d*ln(exp(2*d*x+2*c)+ 2/b*a*exp(d*x+c)-1)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (59) = 118\).
Time = 0.10 (sec) , antiderivative size = 299, normalized size of antiderivative = 5.07 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^{2} d x \cosh \left (d x + c\right )^{2} + a^{2} d x \sinh \left (d x + c\right )^{2} - a^{2} d x + 2 \, a b \cosh \left (d x + c\right ) - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} - b^{2}\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a^{2} d x \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )}{a^{2} b d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right )^{2} - a^{2} b d} \] Input:
integrate(cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas ")
Output:
-(a^2*d*x*cosh(d*x + c)^2 + a^2*d*x*sinh(d*x + c)^2 - a^2*d*x + 2*a*b*cosh (d*x + c) - ((a^2 + b^2)*cosh(d*x + c)^2 + 2*(a^2 + b^2)*cosh(d*x + c)*sin h(d*x + c) + (a^2 + b^2)*sinh(d*x + c)^2 - a^2 - b^2)*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + (b^2*cosh(d*x + c)^2 + 2*b^2*co sh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 - b^2)*log(2*sinh(d*x + c) /(cosh(d*x + c) - sinh(d*x + c))) + 2*(a^2*d*x*cosh(d*x + c) + a*b)*sinh(d *x + c))/(a^2*b*d*cosh(d*x + c)^2 + 2*a^2*b*d*cosh(d*x + c)*sinh(d*x + c) + a^2*b*d*sinh(d*x + c)^2 - a^2*b*d)
\[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate(cosh(d*x+c)*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral(cosh(c + d*x)*coth(c + d*x)**2/(a + b*sinh(c + d*x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.22 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {d x + c}{b d} + \frac {2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} b d} \] Input:
integrate(cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima ")
Output:
(d*x + c)/(b*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(- d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + (a^2 + b^2)*log( -2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*b*d)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (59) = 118\).
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{2} b} - \frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a}{a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}}{d} \] Input:
integrate(cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
-(b*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^2*b) - (b*(e^(d*x + c) - e^(-d*x - c)) - 2*a)/(a^2*(e^(d*x + c) - e^(-d*x - c))))/d
Time = 1.51 (sec) , antiderivative size = 356, normalized size of antiderivative = 6.03 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {x}{b}+\frac {\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{b\,d}+\frac {b\,\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}-\frac {b\,\ln \left (4\,a^6+16\,b^6+32\,a^2\,b^4+20\,a^4\,b^2-4\,a^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-16\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-20\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d} \] Input:
int((cosh(c + d*x)*coth(c + d*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
(2*exp(c + d*x))/(a*d - a*d*exp(2*c + 2*d*x)) - x/b + log(8*a^5*exp(d*x)*e xp(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*d) + (b*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*e xp(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^2*d) - (b*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^ 2*d)
Time = 1.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 4.80 \[ \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{2}+e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}-e^{2 d x +2 c} a^{2} d x -2 e^{d x +c} a b +\mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+\mathrm {log}\left (e^{d x +c}+1\right ) b^{2}-\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{2}-\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}+a^{2} d x}{a^{2} b d \left (e^{2 d x +2 c}-1\right )} \] Input:
int(cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
( - e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b**2 - e**(2*c + 2*d*x)*log(e** (c + d*x) + 1)*b**2 + e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**2 + e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x )*a - b)*b**2 - e**(2*c + 2*d*x)*a**2*d*x - 2*e**(c + d*x)*a*b + log(e**(c + d*x) - 1)*b**2 + log(e**(c + d*x) + 1)*b**2 - log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**2 - log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b )*b**2 + a**2*d*x)/(a**2*b*d*(e**(2*c + 2*d*x) - 1))