Integrand size = 13, antiderivative size = 54 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {a x}{b^2}-\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2}+\frac {\cosh (x)}{b} \]
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 396, normalized size of antiderivative = 7.33 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=\frac {\cosh (x) \left (-2 \sqrt {a-i b} \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right ) \sqrt {1+i \sinh (x)}+2 (a-i b) \text {arctanh}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right ) \sqrt {1+i \sinh (x)}+\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \left (-2 (-1)^{3/4} \sqrt {b} \arcsin \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {b}}\right )+\sqrt {a-i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}\right )\right )}{\sqrt {a-i b} \sqrt {a+i b} b \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}} \]
(Cosh[x]*(-2*Sqrt[a - I*b]*Sqrt[a + I*b]*ArcTanh[Sqrt[-((b*(I + Sinh[x]))/ (a - I*b))]/Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]]*Sqrt[1 + I*Sinh[x]] + 2 *(a - I*b)*ArcTanh[(Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/(S qrt[a + I*b]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))])]*Sqrt[1 + I*Sinh[x]] + Sqrt[a + I*b]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*(-2*(-1)^(3/4)*Sqrt[b ]*ArcSin[((1/2 + I/2)*Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/ Sqrt[b]] + Sqrt[a - I*b]*Sqrt[1 + I*Sinh[x]]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])))/(Sqrt[a - I*b]*Sqrt[a + I*b]*b*Sqrt[1 + I*Sinh[x]]*Sqrt[-((b*(- I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])
Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3174, 26, 3042, 3214, 3042, 3139, 1083, 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.
\(\displaystyle \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)^2}{a-i b \sin (i x)}dx\) |
\(\Big \downarrow \) 3174 |
\(\displaystyle \frac {\cosh (x)}{b}+\frac {i \int -\frac {i (b-a \sinh (x))}{a+b \sinh (x)}dx}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\int \frac {b-a \sinh (x)}{a+b \sinh (x)}dx}{b}+\frac {\cosh (x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (x)}{b}+\frac {\int \frac {b+i a \sin (i x)}{a-i b \sin (i x)}dx}{b}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{b}-\frac {a x}{b}}{b}+\frac {\cosh (x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (x)}{b}+\frac {-\frac {a x}{b}+\frac {\left (a^2+b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{b}}{b}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {\frac {2 \left (a^2+b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{b}-\frac {a x}{b}}{b}+\frac {\cosh (x)}{b}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {-\frac {4 \left (a^2+b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {a x}{b}}{b}+\frac {\cosh (x)}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {-\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}-\frac {a x}{b}}{b}+\frac {\cosh (x)}{b}\) |
(-((a*x)/b) - (2*Sqrt[a^2 + b^2]*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/b)/b + Cosh[x]/b
3.2.92.3.1 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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(b*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Time = 1.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {a x}{b^{2}}+\frac {{\mathrm e}^{x}}{2 b}+\frac {{\mathrm e}^{-x}}{2 b}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{b^{2}}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{b^{2}}\) | \(93\) |
default | \(-\frac {2 \left (-a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}\) | \(100\) |
-a*x/b^2+1/2/b*exp(x)+1/2/b/exp(x)+(a^2+b^2)^(1/2)/b^2*ln(exp(x)-(-a+(a^2+ b^2)^(1/2))/b)-(a^2+b^2)^(1/2)/b^2*ln(exp(x)+(a+(a^2+b^2)^(1/2))/b)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (50) = 100\).
Time = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 3.17 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {2 \, a x \cosh \left (x\right ) - b \cosh \left (x\right )^{2} - b \sinh \left (x\right )^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + 2 \, {\left (a x - b \cosh \left (x\right )\right )} \sinh \left (x\right ) - b}{2 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \]
-1/2*(2*a*x*cosh(x) - b*cosh(x)^2 - b*sinh(x)^2 - 2*sqrt(a^2 + b^2)*(cosh( x) + sinh(x))*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*si nh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*s inh(x) - b)) + 2*(a*x - b*cosh(x))*sinh(x) - b)/(b^2*cosh(x) + b^2*sinh(x) )
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (46) = 92\).
Time = 67.57 (sec) , antiderivative size = 377, normalized size of antiderivative = 6.98 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=\begin {cases} \tilde {\infty } \left (\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}}{b} & \text {for}\: a = 0 \\\frac {- \frac {x \sinh ^{2}{\left (x \right )}}{2} + \frac {x \cosh ^{2}{\left (x \right )}}{2} + \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{2}}{a} & \text {for}\: b = 0 \\- \frac {a x \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {a x}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {2 b}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(log(tanh(x/2))*tanh(x/2)**2/(tanh(x/2)**2 - 1) - log(tanh( x/2))/(tanh(x/2)**2 - 1) - 2/(tanh(x/2)**2 - 1)), Eq(a, 0) & Eq(b, 0)), (( log(tanh(x/2))*tanh(x/2)**2/(tanh(x/2)**2 - 1) - log(tanh(x/2))/(tanh(x/2) **2 - 1) - 2/(tanh(x/2)**2 - 1))/b, Eq(a, 0)), ((-x*sinh(x)**2/2 + x*cosh( x)**2/2 + sinh(x)*cosh(x)/2)/a, Eq(b, 0)), (-a*x*tanh(x/2)**2/(b**2*tanh(x /2)**2 - b**2) + a*x/(b**2*tanh(x/2)**2 - b**2) - 2*b/(b**2*tanh(x/2)**2 - b**2) - sqrt(a**2 + b**2)*log(tanh(x/2) - b/a - sqrt(a**2 + b**2)/a)*tanh (x/2)**2/(b**2*tanh(x/2)**2 - b**2) + sqrt(a**2 + b**2)*log(tanh(x/2) - b/ a - sqrt(a**2 + b**2)/a)/(b**2*tanh(x/2)**2 - b**2) + sqrt(a**2 + b**2)*lo g(tanh(x/2) - b/a + sqrt(a**2 + b**2)/a)*tanh(x/2)**2/(b**2*tanh(x/2)**2 - b**2) - sqrt(a**2 + b**2)*log(tanh(x/2) - b/a + sqrt(a**2 + b**2)/a)/(b** 2*tanh(x/2)**2 - b**2), True))
Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.50 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{2}} \]
-a*x/b^2 + 1/2*e^(-x)/b + 1/2*e^x/b + sqrt(a^2 + b^2)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/b^2
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.54 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{b^{2}} \]
-a*x/b^2 + 1/2*e^(-x)/b + 1/2*e^x/b + sqrt(a^2 + b^2)*log(abs(2*b*e^x + 2* a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/b^2
Time = 1.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.61 \[ \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,b}+\frac {{\mathrm {e}}^x}{2\,b}-\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-b^4}}{b^2\,\sqrt {a^2+b^2}}+\frac {{\mathrm {e}}^x\,\sqrt {-b^4}}{b\,\sqrt {a^2+b^2}}\right )\,\sqrt {a^2+b^2}}{\sqrt {-b^4}}-\frac {a\,x}{b^2} \]