Integrand size = 11, antiderivative size = 72 \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\frac {a b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2} \]
a*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)+a*cosh(x )/(a^2-b^2)-b*sinh(x)/(a^2-b^2)
Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\frac {2 a b \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}+\frac {b \sinh (x)}{-a^2+b^2} \]
(2*a*b*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(3/2) *(a + b)^(3/2)) + (a*Cosh[x])/(a^2 - b^2) + (b*Sinh[x])/(-a^2 + b^2)
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 26, 4001, 26, 3042, 26, 3588, 26, 3042, 26, 3117, 3118, 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.
\(\displaystyle \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x)}{a-i b \tan (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{a-i b \tan (i x)}dx\) |
\(\Big \downarrow \) 4001 |
\(\displaystyle -i \int \frac {i \cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\sinh (x) \cosh (x)}{a \cosh (x)+b \sinh (x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x) \cos (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
\(\Big \downarrow \) 3588 |
\(\displaystyle -i \left (\frac {a \int i \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i a \int \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {i a \int -i \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -i \left (\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -i \left (\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle -i \left (-\frac {a b \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (-\frac {a b \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
(-I)*(-((a*b*ArcTanh[((-I)*b*Cosh[x] - I*a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) + (I*a*Cosh[x])/(a^2 - b^2) - (I*b*Sinh[x])/(a^2 - b^2))
3.1.83.3.1 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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[a*(b/(a^ 2 + b^2)) Int[Cos[c + d*x]^(m - 1)*(Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {2 a b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {a^{2}-b^{2}}}+\frac {4}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(92\) |
risch | \(\frac {{\mathrm e}^{x}}{2 a +2 b}+\frac {{\mathrm e}^{-x}}{2 a -2 b}-\frac {b a \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {b a \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\) | \(120\) |
2*a*b/(a+b)/(a-b)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^ 2)^(1/2))+4/(4*a-4*b)/(tanh(1/2*x)+1)-4/(4*a+4*b)/(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (68) = 136\).
Time = 0.27 (sec) , antiderivative size = 427, normalized size of antiderivative = 5.93 \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\left [\frac {a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, \frac {a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 4 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\right ] \]
[1/2*(a^3 + a^2*b - a*b^2 - b^3 + (a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2 + 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)*sinh(x) + (a^3 - a^2*b - a*b^2 + b^3 )*sinh(x)^2 + 2*(a*b*cosh(x) + a*b*sinh(x))*sqrt(-a^2 + b^2)*log(((a + b)* cosh(x)^2 + 2*(a + b)*cosh(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)*s inh(x) + (a + b)*sinh(x)^2 + a - b)))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x) + ( a^4 - 2*a^2*b^2 + b^4)*sinh(x)), 1/2*(a^3 + a^2*b - a*b^2 - b^3 + (a^3 - a ^2*b - a*b^2 + b^3)*cosh(x)^2 + 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)*sinh (x) + (a^3 - a^2*b - a*b^2 + b^3)*sinh(x)^2 - 4*(a*b*cosh(x) + a*b*sinh(x) )*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x ))))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x))]
\[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\int \frac {\sinh {\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
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 = 60, normalized size of antiderivative = 0.83 \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\frac {2 \, a b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} + \frac {e^{\left (-x\right )}}{2 \, {\left (a - b\right )}} + \frac {e^{x}}{2 \, {\left (a + b\right )}} \]
2*a*b*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(a^2 - b^2)^(3/2) + 1/2*e^(- x)/(a - b) + 1/2*e^x/(a + b)
Time = 1.73 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.18 \[ \int \frac {\sinh (x)}{a+b \tanh (x)} \, dx=\frac {{\mathrm {e}}^x}{2\,a+2\,b}+\frac {{\mathrm {e}}^{-x}}{2\,a-2\,b}+\frac {2\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^x\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{a^3\,\sqrt {a^2\,b^2}+b^3\,\sqrt {a^2\,b^2}-a\,b^2\,\sqrt {a^2\,b^2}-a^2\,b\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}} \]