Integrand size = 16, antiderivative size = 68 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}-\frac {a}{\left (a^2-b^2\right ) (b+a \coth (x))}-\frac {2 a b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \] Output:
(a^2+b^2)*x/(a^2-b^2)^2-a/(a^2-b^2)/(b+a*coth(x))-2*a*b*ln(a*cosh(x)+b*sin h(x))/(a^2-b^2)^2
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\left (a^2+b^2\right ) x-2 a b \log (a \cosh (x)+b \sinh (x))-\frac {a (a-b) (a+b) \sinh (x)}{a \cosh (x)+b \sinh (x)}}{(a-b)^2 (a+b)^2} \] Input:
Integrate[Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2,x]
Output:
((a^2 + b^2)*x - 2*a*b*Log[a*Cosh[x] + b*Sinh[x]] - (a*(a - b)*(a + b)*Sin h[x])/(a*Cosh[x] + b*Sinh[x]))/((a - b)^2*(a + b)^2)
Time = 0.90 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 25, 3564, 3042, 3964, 3042, 4014, 26, 3042, 4013}
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.
| \(\displaystyle \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3564 |
| \(\displaystyle -\int \frac {1}{(-i b-i a \coth (x))^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {1}{\left (-i b-a \tan \left (i x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle -\frac {\int \frac {b-a \coth (x)}{b+a \coth (x)}dx}{a^2-b^2}-\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {\int \frac {b+i a \tan \left (i x+\frac {\pi }{2}\right )}{b-i a \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {-\frac {x \left (a^2+b^2\right )}{a^2-b^2}+\frac {2 i a b \int -\frac {i (a+b \coth (x))}{b+a \coth (x)}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {2 a b \int \frac {a+b \coth (x)}{b+a \coth (x)}dx}{a^2-b^2}-\frac {x \left (a^2+b^2\right )}{a^2-b^2}}{a^2-b^2}-\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {-\frac {x \left (a^2+b^2\right )}{a^2-b^2}+\frac {2 a b \int \frac {a-i b \tan \left (i x+\frac {\pi }{2}\right )}{b-i a \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {\frac {2 a b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}-\frac {x \left (a^2+b^2\right )}{a^2-b^2}}{a^2-b^2}\) |
Input:
Int[Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2,x]
Output:
-(a/((a^2 - b^2)*(b + a*Coth[x]))) - (-(((a^2 + b^2)*x)/(a^2 - b^2)) + (2* a*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2))/(a^2 - b^2)
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[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0 ]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{2}}+\frac {2 a \left (\frac {\left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a +2 b \tanh \left (\frac {x}{2}\right )+a \tanh \left (\frac {x}{2}\right )^{2}}-b \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \tanh \left (\frac {x}{2}\right )^{2}\right )\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{2}}\) | \(101\) |
| parallelrisch | \(\frac {\left (-2 \tanh \left (x \right ) a \,b^{3}-2 b^{2} a^{2}\right ) \ln \left (a +b \tanh \left (x \right )\right )+\left (2 \tanh \left (x \right ) a \,b^{3}+2 b^{2} a^{2}\right ) \ln \left (1-\tanh \left (x \right )\right )+\left (a +b \right ) \left (x \,b^{2} \left (a +b \right ) \tanh \left (x \right )+a \left (b^{2} x +a \left (-1+x \right ) b +a^{2}\right )\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \left (a +b \tanh \left (x \right )\right ) b}\) | \(108\) |
| risch | \(\frac {x}{a^{2}+2 a b +b^{2}}+\frac {4 a b x}{a^{4}-2 b^{2} a^{2}+b^{4}}+\frac {2 a^{2}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4}-2 b^{2} a^{2}+b^{4}}\) | \(121\) |
Input:
int(sinh(x)^2/(a*cosh(x)+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-1/(a+b)^2*ln(tanh(1/2*x)-1)+2*a/(a-b)^2/(a+b)^2*((-a^2+b^2)*tanh(1/2*x)/( a+2*b*tanh(1/2*x)+a*tanh(1/2*x)^2)-b*ln(a+2*b*tanh(1/2*x)+a*tanh(1/2*x)^2) )+1/(a-b)^2*ln(1+tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (68) = 136\).
Time = 0.11 (sec) , antiderivative size = 348, normalized size of antiderivative = 5.12 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{3} - 2 \, a^{2} b + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} x - 2 \, {\left (a^{2} b - a b^{2} + {\left (a^{2} b + a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} b + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sinh \left (x\right )^{2}} \] Input:
integrate(sinh(x)^2/(a*cosh(x)+b*sinh(x))^2,x, algorithm="fricas")
Output:
((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x*cosh(x)^2 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x*cosh(x)*sinh(x) + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x*sinh(x)^2 + 2 *a^3 - 2*a^2*b + (a^3 + a^2*b - a*b^2 - b^3)*x - 2*(a^2*b - a*b^2 + (a^2*b + a*b^2)*cosh(x)^2 + 2*(a^2*b + a*b^2)*cosh(x)*sinh(x) + (a^2*b + a*b^2)* sinh(x)^2)*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))))/(a^5 - a^4* b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2 *b^3 + a*b^4 + b^5)*cosh(x)^2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a *b^4 + b^5)*cosh(x)*sinh(x) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*sinh(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (56) = 112\).
Time = 0.74 (sec) , antiderivative size = 945, normalized size of antiderivative = 13.90 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(x)**2/(a*cosh(x)+b*sinh(x))**2,x)
Output:
Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), ((x - sinh(x)/cosh(x))/a**2, Eq(b, 0)), (2*x*sinh(x)**2/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*cosh(x) + 8*b** 2*cosh(x)**2) - 4*x*sinh(x)*cosh(x)/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*c osh(x) + 8*b**2*cosh(x)**2) + 2*x*cosh(x)**2/(8*b**2*sinh(x)**2 - 16*b**2* sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + 3*sinh(x)**2/(8*b**2*sinh(x)**2 - 1 6*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) - cosh(x)**2/(8*b**2*sinh(x)** 2 - 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2), Eq(a, -b)), (2*x*sinh(x) **2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + 4* x*sinh(x)*cosh(x)/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*co sh(x)**2) + 2*x*cosh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) - 3*sinh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cos h(x) + 8*b**2*cosh(x)**2) + cosh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x )*cosh(x) + 8*b**2*cosh(x)**2), Eq(a, b)), (a**3*x*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh( x) + b**5*sinh(x)) - a**3*sinh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3* b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) + a**2 *b*x*sinh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2 *b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) - 2*a**2*b*log(a*cosh(x)/b + sinh(x))*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) + a*b**2*x*cosh(x)...
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2 \, a b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, a^{2}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{a^{2} + 2 \, a b + b^{2}} \] Input:
integrate(sinh(x)^2/(a*cosh(x)+b*sinh(x))^2,x, algorithm="maxima")
Output:
-2*a*b*log(-(a - b)*e^(-2*x) - a - b)/(a^4 - 2*a^2*b^2 + b^4) - 2*a^2/(a^4 - 2*a^2*b^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*x)) + x/(a^2 + 2*a*b + b^2)
Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.66 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2 \, a b \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {x}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, {\left (a b e^{\left (2 \, x\right )} + a^{2} - a b\right )}}{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \] Input:
integrate(sinh(x)^2/(a*cosh(x)+b*sinh(x))^2,x, algorithm="giac")
Output:
-2*a*b*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^4 - 2*a^2*b^2 + b^4) + x /(a^2 - 2*a*b + b^2) + 2*(a*b*e^(2*x) + a^2 - a*b)/((a^3 - a^2*b - a*b^2 + b^3)*(a*e^(2*x) + b*e^(2*x) + a - b))
Time = 0.52 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.59 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\frac {a^2\,\mathrm {cosh}\left (x\right )}{b\,\left (a^2-b^2\right )}+\frac {a\,x\,\mathrm {cosh}\left (x\right )\,\left (a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}+\frac {b\,x\,\mathrm {sinh}\left (x\right )\,\left (a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}}{a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )}+\ln \left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^2}-\frac {1}{2\,{\left (a-b\right )}^2}\right ) \] Input:
int(sinh(x)^2/(a*cosh(x) + b*sinh(x))^2,x)
Output:
((a^2*cosh(x))/(b*(a^2 - b^2)) + (a*x*cosh(x)*(a^2 + b^2))/(a^2 - b^2)^2 + (b*x*sinh(x)*(a^2 + b^2))/(a^2 - b^2)^2)/(a*cosh(x) + b*sinh(x)) + log(a* cosh(x) + b*sinh(x))*(1/(2*(a + b)^2) - 1/(2*(a - b)^2))
Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.07 \[ \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {-2 \cosh \left (x \right ) \mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) a^{2} b^{2}+\cosh \left (x \right ) a^{4}+\cosh \left (x \right ) a^{3} b x -\cosh \left (x \right ) a^{2} b^{2}+\cosh \left (x \right ) a \,b^{3} x -2 \,\mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) \sinh \left (x \right ) a \,b^{3}+\sinh \left (x \right ) a^{2} b^{2} x +\sinh \left (x \right ) b^{4} x}{b \left (\cosh \left (x \right ) a^{5}-2 \cosh \left (x \right ) a^{3} b^{2}+\cosh \left (x \right ) a \,b^{4}+\sinh \left (x \right ) a^{4} b -2 \sinh \left (x \right ) a^{2} b^{3}+\sinh \left (x \right ) b^{5}\right )} \] Input:
int(sinh(x)^2/(a*cosh(x)+b*sinh(x))^2,x)
Output:
( - 2*cosh(x)*log(cosh(x)*a + sinh(x)*b)*a**2*b**2 + cosh(x)*a**4 + cosh(x )*a**3*b*x - cosh(x)*a**2*b**2 + cosh(x)*a*b**3*x - 2*log(cosh(x)*a + sinh (x)*b)*sinh(x)*a*b**3 + sinh(x)*a**2*b**2*x + sinh(x)*b**4*x)/(b*(cosh(x)* a**5 - 2*cosh(x)*a**3*b**2 + cosh(x)*a*b**4 + sinh(x)*a**4*b - 2*sinh(x)*a **2*b**3 + sinh(x)*b**5))