Integrand size = 14, antiderivative size = 66 \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))} \]
-b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)-a/(a^2-b^ 2)/(a*cosh(x)+b*sinh(x))
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.89 \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {a \sqrt {a-b} (a+b)+2 a b \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \cosh (x)+2 b^2 \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sinh (x)}{(a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))} \]
-((a*Sqrt[a - b]*(a + b) + 2*a*b*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqr t[a - b]*Sqrt[a + b])]*Cosh[x] + 2*b^2*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2] )/(Sqrt[a - b]*Sqrt[a + b])]*Sinh[x])/((a - b)^(3/2)*(a + b)^2*(a*Cosh[x] + b*Sinh[x])))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 26, 3633, 3042, 3553, 219}
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.
\(\displaystyle \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x)}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 3633 |
\(\displaystyle -i \left (-\frac {i b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (-\frac {i b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\right )\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle -i \left (\frac {b \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}-\frac {i a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (\frac {b \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {i a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\right )\) |
(-I)*((b*ArcTanh[((-I)*b*Cosh[x] - I*a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b ^2)^(3/2) - (I*a)/((a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])))
3.7.96.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[((A_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_) ]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[-(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Simp[(a*A - c*C)/(a^2 - b^2 - c^2) Int[1/( a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, C} , x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - c*C, 0]
Time = 1.66 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {-8 b \tanh \left (\frac {x}{2}\right )-8 a}{\left (4 a^{2}-4 b^{2}\right ) \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}-\frac {8 b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (4 a^{2}-4 b^{2}\right ) \sqrt {a^{2}-b^{2}}}\) | \(99\) |
risch | \(-\frac {2 a \,{\mathrm e}^{x}}{\left (a -b \right ) \left (a +b \right ) \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {b \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\) | \(132\) |
4*(-2*b*tanh(1/2*x)-2*a)/(4*a^2-4*b^2)/(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a) -8*b/(4*a^2-4*b^2)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b ^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (62) = 124\).
Time = 0.26 (sec) , antiderivative size = 594, normalized size of antiderivative = 9.00 \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\left [\frac {{\left ({\left (a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a b + b^{2}\right )} \sinh \left (x\right )^{2} + a b - b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right ) - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} - a b^{2}\right )} \sinh \left (x\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}}, \frac {2 \, {\left ({\left ({\left (a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a b + b^{2}\right )} \sinh \left (x\right )^{2} + a b - b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right ) - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) - {\left (a^{3} - a b^{2}\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}}\right ] \]
[(((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sin h(x)^2 + a*b - b^2)*sqrt(-a^2 + b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b)*co sh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x) ^2 + a - b)) - 2*(a^3 - a*b^2)*cosh(x) - 2*(a^3 - a*b^2)*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), 2*(((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*s inh(x) + (a*b + b^2)*sinh(x)^2 + a*b - b^2)*sqrt(a^2 - b^2)*arctan(sqrt(a^ 2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))) - (a^3 - a*b^2)*cosh(x) - (a ^3 - a*b^2)*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)]
Timed out. \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2 \, b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, a e^{x}}{{\left (a^{2} - b^{2}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \]
-2*b*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(a^2 - b^2)^(3/2) - 2*a*e^x/( (a^2 - b^2)*(a*e^(2*x) + b*e^(2*x) + a - b))
Time = 2.58 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.77 \[ \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (b^2\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+a\,b\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}\right )}{a^4\,\sqrt {b^2}-2\,a^2\,{\left (b^2\right )}^{3/2}+b^4\,\sqrt {b^2}+a\,b\,{\left (b^2\right )}^{3/2}-a\,b^3\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}-\frac {2\,a\,{\mathrm {e}}^x}{\left (a+b\right )\,\left (a-b\right )\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
- (2*atan((exp(x)*(b^2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2) + a*b*(a^ 6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2)))/(a^4*(b^2)^(1/2) - 2*a^2*(b^2)^(3 /2) + b^4*(b^2)^(1/2) + a*b*(b^2)^(3/2) - a*b^3*(b^2)^(1/2)))*(b^2)^(1/2)) /(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2) - (2*a*exp(x))/((a + b)*(a - b) *(a - b + exp(2*x)*(a + b)))