Integrand size = 15, antiderivative size = 99 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\frac {2 A \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (1-\cosh (x))}{2 (a+b)}-\frac {B \log (1+\cosh (x))}{2 (a-b)}+\frac {b B \log (a+b \cosh (x))}{a^2-b^2} \] Output:
2*A*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(1/2)/(a+b)^(1/2)+B *ln(1-cosh(x))/(2*a+2*b)-B*ln(1+cosh(x))/(2*a-2*b)+b*B*ln(a+b*cosh(x))/(a^ 2-b^2)
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\frac {-2 A \left (a^2-b^2\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )-\sqrt {-a^2+b^2} B \left ((a+b) \log \left (\cosh \left (\frac {x}{2}\right )\right )-b \log (a+b \cosh (x))+(-a+b) \log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{(a-b) (a+b) \sqrt {-a^2+b^2}} \] Input:
Integrate[(A + B*Csch[x])/(a + b*Cosh[x]),x]
Output:
(-2*A*(a^2 - b^2)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]] - Sqrt[-a^2 + b^2]*B*((a + b)*Log[Cosh[x/2]] - b*Log[a + b*Cosh[x]] + (-a + b)*Log[Si nh[x/2]]))/((a - b)*(a + b)*Sqrt[-a^2 + b^2])
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 4713, 26, 26, 3042, 26, 4901, 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.
\(\displaystyle \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+i B \csc (i x)}{a+b \cos (i x)}dx\) |
\(\Big \downarrow \) 4713 |
\(\displaystyle \int -\frac {i \text {csch}(x) (i A \sinh (x)+i B)}{a+b \cosh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {i \text {csch}(x) (B+A \sinh (x))}{a+b \cosh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\text {csch}(x) (A \sinh (x)+B)}{a+b \cosh (x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i (B-i A \sin (i x))}{\sin (i x) (a+b \cos (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {B-i A \sin (i x)}{(a+b \cos (i x)) \sin (i x)}dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle i \int \left (-\frac {i A}{a+b \cosh (x)}-\frac {i B \text {csch}(x)}{a+b \cosh (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {i b B \log (a+b \cosh (x))}{a^2-b^2}-\frac {2 i A \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {i B \log (1-\cosh (x))}{2 (a+b)}+\frac {i B \log (\cosh (x)+1)}{2 (a-b)}\right )\) |
Input:
Int[(A + B*Csch[x])/(a + b*Cosh[x]),x]
Output:
I*(((-2*I)*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sq rt[a + b]) - ((I/2)*B*Log[1 - Cosh[x]])/(a + b) + ((I/2)*B*Log[1 + Cosh[x] ])/(a - b) - (I*b*B*Log[a + b*Cosh[x]])/(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[(csc[(a_.) + (b_.)*(x_)]*(B_.) + (A_))*(u_), x_Symbol] :> Int[ActivateT rig[u]*((B + A*Sin[a + b*x])/Sin[a + b*x]), x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 1.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {\frac {B b \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-a -b \right )}{a -b}-\frac {\left (-2 A a -2 A b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{a +b}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a +b}\) | \(100\) |
risch | \(\frac {x B}{a -b}-\frac {x B}{a +b}+\frac {2 x B \,a^{2} b}{-a^{4}+2 a^{2} b^{2}-b^{4}}-\frac {2 x B \,b^{3}}{-a^{4}+2 a^{2} b^{2}-b^{4}}-\frac {B \ln \left ({\mathrm e}^{x}+1\right )}{a -b}+\frac {B \ln \left ({\mathrm e}^{x}-1\right )}{a +b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {A a -\sqrt {A^{2} a^{2}-A^{2} b^{2}}}{A b}\right ) B b}{\left (a +b \right ) \left (a -b \right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {A a -\sqrt {A^{2} a^{2}-A^{2} b^{2}}}{A b}\right ) \sqrt {A^{2} a^{2}-A^{2} b^{2}}}{\left (a +b \right ) \left (a -b \right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {A a +\sqrt {A^{2} a^{2}-A^{2} b^{2}}}{A b}\right ) B b}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left ({\mathrm e}^{x}+\frac {A a +\sqrt {A^{2} a^{2}-A^{2} b^{2}}}{A b}\right ) \sqrt {A^{2} a^{2}-A^{2} b^{2}}}{\left (a +b \right ) \left (a -b \right )}\) | \(334\) |
Input:
int((A+B*csch(x))/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
Output:
1/(a+b)*(B*b/(a-b)*ln(tanh(1/2*x)^2*a-b*tanh(1/2*x)^2-a-b)-(-2*A*a-2*A*b)/ ((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2)))+B/(a+b )*ln(tanh(1/2*x))
Time = 1.75 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.01 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\left [\frac {B b \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \sqrt {a^{2} - b^{2}} A \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - {\left (B a + B b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (B a - B b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} - b^{2}}, \frac {B b \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, \sqrt {-a^{2} + b^{2}} A \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) - {\left (B a + B b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (B a - B b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} - b^{2}}\right ] \] Input:
integrate((A+B*csch(x))/(a+b*cosh(x)),x, algorithm="fricas")
Output:
[(B*b*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) + sqrt(a^2 - b^2)*A*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*sinh(x) + a))/(b*cosh (x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) + b)) - (B*a + B*b)*log(cosh(x) + sinh(x) + 1) + (B*a - B*b)*log(cosh(x) + sinh(x) - 1 ))/(a^2 - b^2), (B*b*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) - 2*sqrt(- a^2 + b^2)*A*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b ^2)) - (B*a + B*b)*log(cosh(x) + sinh(x) + 1) + (B*a - B*b)*log(cosh(x) + sinh(x) - 1))/(a^2 - b^2)]
\[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\int \frac {A + B \operatorname {csch}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \] Input:
integrate((A+B*csch(x))/(a+b*cosh(x)),x)
Output:
Integral((A + B*csch(x))/(a + b*cosh(x)), x)
Exception generated. \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*csch(x))/(a+b*cosh(x)),x, algorithm="maxima")
Output:
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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\frac {B b \log \left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}{a^{2} - b^{2}} + \frac {2 \, A \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} - \frac {B \log \left (e^{x} + 1\right )}{a - b} + \frac {B \log \left ({\left | e^{x} - 1 \right |}\right )}{a + b} \] Input:
integrate((A+B*csch(x))/(a+b*cosh(x)),x, algorithm="giac")
Output:
B*b*log(b*e^(2*x) + 2*a*e^x + b)/(a^2 - b^2) + 2*A*arctan((b*e^x + a)/sqrt (-a^2 + b^2))/sqrt(-a^2 + b^2) - B*log(e^x + 1)/(a - b) + B*log(abs(e^x - 1))/(a + b)
Time = 4.47 (sec) , antiderivative size = 983, normalized size of antiderivative = 9.93 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx =\text {Too large to display} \] Input:
int((A + B/sinh(x))/(a + b*cosh(x)),x)
Output:
(log((((32*(A^2*b^3 + A^2*a^2*b + 2*B^2*a^2*b + 4*B^2*a^3*exp(x) + 3*B^2*a *b^2*exp(x) - 4*A*B*a*b^2 + 2*A*B*b^3*exp(x) + 2*A^2*a*b^2*exp(x) - 8*A*B* a^2*b*exp(x)))/b^5 + (((32*(2*B*b^4 + B*a^2*b^2 - 4*A*a^4*exp(x) - A*b^4*e xp(x) + 2*A*a*b^3 - 2*A*a^3*b + 6*B*a*b^3*exp(x) - 3*B*a^3*b*exp(x) + 5*A* a^2*b^2*exp(x)))/b^5 - (32*(A*((a + b)^3*(a - b)^3)^(1/2) - B*b^3 + B*a^2* b)*(3*a^4*b - 3*a^2*b^3 + 4*a^5*exp(x) + a*b^4*exp(x) - 5*a^3*b^2*exp(x))) /(b^5*(a^4 + b^4 - 2*a^2*b^2)))*(A*((a + b)^3*(a - b)^3)^(1/2) - B*b^3 + B *a^2*b))/(a^4 + b^4 - 2*a^2*b^2))*(A*((a + b)^3*(a - b)^3)^(1/2) - B*b^3 + B*a^2*b))/(a^4 + b^4 - 2*a^2*b^2) - (32*(2*B^3*b^2 + A^2*B*b^2 - 2*A*B^2* a*b + 4*B^3*a*b*exp(x) - 4*A*B^2*a^2*exp(x) + A*B^2*b^2*exp(x) + A^2*B*a*b *exp(x)))/b^5)*(A*((a + b)^3*(a - b)^3)^(1/2) - B*b^3 + B*a^2*b))/(a^4 + b ^4 - 2*a^2*b^2) - (B*log(exp(x) + 1))/(a - b) - (log(- (32*(2*B^3*b^2 + A^ 2*B*b^2 - 2*A*B^2*a*b + 4*B^3*a*b*exp(x) - 4*A*B^2*a^2*exp(x) + A*B^2*b^2* exp(x) + A^2*B*a*b*exp(x)))/b^5 - (((32*(A^2*b^3 + A^2*a^2*b + 2*B^2*a^2*b + 4*B^2*a^3*exp(x) + 3*B^2*a*b^2*exp(x) - 4*A*B*a*b^2 + 2*A*B*b^3*exp(x) + 2*A^2*a*b^2*exp(x) - 8*A*B*a^2*b*exp(x)))/b^5 - (((32*(2*B*b^4 + B*a^2*b ^2 - 4*A*a^4*exp(x) - A*b^4*exp(x) + 2*A*a*b^3 - 2*A*a^3*b + 6*B*a*b^3*exp (x) - 3*B*a^3*b*exp(x) + 5*A*a^2*b^2*exp(x)))/b^5 + (32*(B*b^3 + A*((a + b )^3*(a - b)^3)^(1/2) - B*a^2*b)*(3*a^4*b - 3*a^2*b^3 + 4*a^5*exp(x) + a*b^ 4*exp(x) - 5*a^3*b^2*exp(x)))/(b^5*(a^4 + b^4 - 2*a^2*b^2)))*(B*b^3 + A...
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \text {csch}(x)}{a+b \cosh (x)} \, dx=\frac {-2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a +\mathrm {log}\left (e^{x}-1\right ) a b -\mathrm {log}\left (e^{x}-1\right ) b^{2}-\mathrm {log}\left (e^{x}+1\right ) a b -\mathrm {log}\left (e^{x}+1\right ) b^{2}+\mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) b^{2}}{a^{2}-b^{2}} \] Input:
int((A+B*csch(x))/(a+b*cosh(x)),x)
Output:
( - 2*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a + log (e**x - 1)*a*b - log(e**x - 1)*b**2 - log(e**x + 1)*a*b - log(e**x + 1)*b* *2 + log(e**(2*x)*b + 2*e**x*a + b)*b**2)/(a**2 - b**2)