Integrand size = 12, antiderivative size = 107 \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\frac {a x}{a^2-b^2}-\frac {2 a c \arctan \left (\frac {b+(a-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2-c^2}}-\frac {b \log (c+a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
a*x/(a^2-b^2)-b*ln(c+a*cosh(x)+b*sinh(x))/(a^2-b^2)-2*a*c*arctan((b+(a-c)* tanh(1/2*x))/(a^2-b^2-c^2)^(1/2))/(a^2-b^2)/(a^2-b^2-c^2)^(1/2)
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\frac {a x-\frac {2 a c \arctan \left (\frac {b+(a-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}}-b \log (c+a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
(a*x - (2*a*c*ArcTan[(b + (a - c)*Tanh[x/2])/Sqrt[a^2 - b^2 - c^2]])/Sqrt[ a^2 - b^2 - c^2] - b*Log[c + a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3638, 3042, 3617, 3042, 3603, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{a+b \tanh (x)+c \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-i b \tan (i x)+c \sec (i x)}dx\) |
| \(\Big \downarrow \) 3638 |
| \(\displaystyle \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)+c}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)+c}dx\) |
| \(\Big \downarrow \) 3617 |
| \(\displaystyle -\frac {a c \int \frac {1}{c+a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x)+c)}{a^2-b^2}+\frac {a x}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a c \int \frac {1}{c+a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x)+c)}{a^2-b^2}+\frac {a x}{a^2-b^2}\) |
| \(\Big \downarrow \) 3603 |
| \(\displaystyle -\frac {2 a c \int \frac {1}{(a-c) \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a+c}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x)+c)}{a^2-b^2}+\frac {a x}{a^2-b^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {4 a c \int \frac {1}{-\left (2 b+2 (a-c) \tanh \left (\frac {x}{2}\right )\right )^2-4 \left (a^2-b^2-c^2\right )}d\left (2 b+2 (a-c) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x)+c)}{a^2-b^2}+\frac {a x}{a^2-b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 a c \arctan \left (\frac {2 (a-c) \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2-c^2}}-\frac {b \log (a \cosh (x)+b \sinh (x)+c)}{a^2-b^2}+\frac {a x}{a^2-b^2}\) |
(a*x)/(a^2 - b^2) - (2*a*c*ArcTan[(2*b + 2*(a - c)*Tanh[x/2])/(2*Sqrt[a^2 - b^2 - c^2])])/((a^2 - b^2)*Sqrt[a^2 - b^2 - c^2]) - (b*Log[c + a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)
3.8.80.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-1), x_Symbol] :> Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, Simp[2*(f /e) Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d + e*x) /2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.))/((a_.) + cos[(d_.) + (e_.)*(x_) ]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[b*B*((d + e*x)/ (e*(b^2 + c^2))), x] + (Simp[c*B*(Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/ (e*(b^2 + c^2))), x] + Simp[(A*(b^2 + c^2) - a*b*B)/(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, B}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*b*B, 0]
Int[((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)]) ^(-1), x_Symbol] :> Int[Cos[d + e*x]/(b + a*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.43 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.65
| method | result | size |
| default | \(\frac {2 \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{2 a -2 b}-\frac {2 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a +2 b}+\frac {\frac {2 \left (-a b +b c \right ) \ln \left (a \tanh \left (\frac {x}{2}\right )^{2}-c \tanh \left (\frac {x}{2}\right )^{2}+2 b \tanh \left (\frac {x}{2}\right )+a +c \right )}{2 a -2 c}+\frac {2 \left (-a c -b^{2}-\frac {\left (-a b +b c \right ) b}{a -c}\right ) \arctan \left (\frac {2 \left (a -c \right ) \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{\sqrt {a^{2}-b^{2}-c^{2}}}}{\left (a -b \right ) \left (a +b \right )}\) | \(177\) |
| risch | \(\frac {x}{a +b}+\frac {2 x \,a^{2} b}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}-\frac {2 x \,b^{3}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}-\frac {2 x b \,c^{2}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {-a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{4}-2 a^{2} b^{2}-a^{2} c^{2}+b^{4}+b^{2} c^{2}}\) | \(878\) |
2/(2*a-2*b)*ln(1+tanh(1/2*x))-2/(2*a+2*b)*ln(tanh(1/2*x)-1)+2/(a-b)/(a+b)* (1/2*(-a*b+b*c)/(a-c)*ln(a*tanh(1/2*x)^2-c*tanh(1/2*x)^2+2*b*tanh(1/2*x)+a +c)+(-a*c-b^2-(-a*b+b*c)*b/(a-c))/(a^2-b^2-c^2)^(1/2)*arctan(1/2*(2*(a-c)* tanh(1/2*x)+2*b)/(a^2-b^2-c^2)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 429, normalized size of antiderivative = 4.01 \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\left [\frac {\sqrt {-a^{2} + b^{2} + c^{2}} a c \log \left (\frac {2 \, {\left (a + b\right )} c \cosh \left (x\right ) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} + b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a + b\right )} c + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 2 \, \sqrt {-a^{2} + b^{2} + c^{2}} {\left ({\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right ) + c\right )}}{{\left (a + b\right )} \cosh \left (x\right )^{2} + {\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \, {\left ({\left (a + b\right )} \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) + a - b}\right ) + {\left (a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a + b\right )} c^{2}\right )} x - {\left (a^{2} b - b^{3} - b c^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{2} - b^{2}\right )} c^{2}}, \frac {2 \, \sqrt {a^{2} - b^{2} - c^{2}} a c \arctan \left (-\frac {{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right ) + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right ) + {\left (a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a + b\right )} c^{2}\right )} x - {\left (a^{2} b - b^{3} - b c^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{2} - b^{2}\right )} c^{2}}\right ] \]
[(sqrt(-a^2 + b^2 + c^2)*a*c*log((2*(a + b)*c*cosh(x) + (a^2 + 2*a*b + b^2 )*cosh(x)^2 + (a^2 + 2*a*b + b^2)*sinh(x)^2 - a^2 + b^2 + 2*c^2 + 2*((a + b)*c + (a^2 + 2*a*b + b^2)*cosh(x))*sinh(x) - 2*sqrt(-a^2 + b^2 + c^2)*((a + b)*cosh(x) + (a + b)*sinh(x) + c))/((a + b)*cosh(x)^2 + (a + b)*sinh(x) ^2 + 2*c*cosh(x) + 2*((a + b)*cosh(x) + c)*sinh(x) + a - b)) + (a^3 + a^2* b - a*b^2 - b^3 - (a + b)*c^2)*x - (a^2*b - b^3 - b*c^2)*log(2*(a*cosh(x) + b*sinh(x) + c)/(cosh(x) - sinh(x))))/(a^4 - 2*a^2*b^2 + b^4 - (a^2 - b^2 )*c^2), (2*sqrt(a^2 - b^2 - c^2)*a*c*arctan(-((a + b)*cosh(x) + (a + b)*si nh(x) + c)/sqrt(a^2 - b^2 - c^2)) + (a^3 + a^2*b - a*b^2 - b^3 - (a + b)*c ^2)*x - (a^2*b - b^3 - b*c^2)*log(2*(a*cosh(x) + b*sinh(x) + c)/(cosh(x) - sinh(x))))/(a^4 - 2*a^2*b^2 + b^4 - (a^2 - b^2)*c^2)]
\[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\int \frac {1}{a + b \tanh {\left (x \right )} + c \operatorname {sech}{\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c^2+b^2-a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99 \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=-\frac {2 \, a c \arctan \left (\frac {a e^{x} + b e^{x} + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )}{\sqrt {a^{2} - b^{2} - c^{2}} {\left (a^{2} - b^{2}\right )}} - \frac {b \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + 2 \, c e^{x} + a - b\right )}{a^{2} - b^{2}} + \frac {x}{a - b} \]
-2*a*c*arctan((a*e^x + b*e^x + c)/sqrt(a^2 - b^2 - c^2))/(sqrt(a^2 - b^2 - c^2)*(a^2 - b^2)) - b*log(a*e^(2*x) + b*e^(2*x) + 2*c*e^x + a - b)/(a^2 - b^2) + x/(a - b)
Time = 7.25 (sec) , antiderivative size = 472, normalized size of antiderivative = 4.41 \[ \int \frac {1}{a+c \text {sech}(x)+b \tanh (x)} \, dx=\frac {x}{a-b}+\frac {\ln \left (a-b+2\,c\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (-2\,a^2\,b+2\,b^3+2\,b\,c^2\right )}{2\,\left (a^4-2\,a^2\,b^2-a^2\,c^2+b^4+b^2\,c^2\right )}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,a\,c}{{\left (a+b\right )}^2\,\left (a^2-b^2\right )\,{\left (a-b\right )}^2\,\sqrt {a^2\,c^2}}-\frac {2\,\left (a^2\,c\,\sqrt {a^2\,c^2}-b^2\,c\,\sqrt {a^2\,c^2}\right )}{a\,{\left (a+b\right )}^2\,{\left (a^2-b^2\right )}^2\,{\left (a-b\right )}^2\,\left (-a^2+b^2+c^2\right )}\right )-\frac {2\,\left (a^3\,\sqrt {a^2\,c^2}+b^3\,\sqrt {a^2\,c^2}-a\,b^2\,\sqrt {a^2\,c^2}-a^2\,b\,\sqrt {a^2\,c^2}\right )}{a\,{\left (a+b\right )}^2\,{\left (a^2-b^2\right )}^2\,{\left (a-b\right )}^2\,\left (-a^2+b^2+c^2\right )}\right )\,\left (\frac {a^3\,\sqrt {-{\left (a^2-b^2\right )}^2\,\left (-a^2+b^2+c^2\right )}}{2}-\frac {b^3\,\sqrt {-{\left (a^2-b^2\right )}^2\,\left (-a^2+b^2+c^2\right )}}{2}-\frac {a\,b^2\,\sqrt {-{\left (a^2-b^2\right )}^2\,\left (-a^2+b^2+c^2\right )}}{2}+\frac {a^2\,b\,\sqrt {-{\left (a^2-b^2\right )}^2\,\left (-a^2+b^2+c^2\right )}}{2}\right )\right )\,\sqrt {a^2\,c^2}}{\sqrt {-{\left (a^2-b^2\right )}^2\,\left (-a^2+b^2+c^2\right )}} \]
x/(a - b) + (log(a - b + 2*c*exp(x) + a*exp(2*x) + b*exp(2*x))*(2*b*c^2 - 2*a^2*b + 2*b^3))/(2*(a^4 + b^4 - 2*a^2*b^2 - a^2*c^2 + b^2*c^2)) - (2*ata n((exp(x)*((2*a*c)/((a + b)^2*(a^2 - b^2)*(a - b)^2*(a^2*c^2)^(1/2)) - (2* (a^2*c*(a^2*c^2)^(1/2) - b^2*c*(a^2*c^2)^(1/2)))/(a*(a + b)^2*(a^2 - b^2)^ 2*(a - b)^2*(b^2 - a^2 + c^2))) - (2*(a^3*(a^2*c^2)^(1/2) + b^3*(a^2*c^2)^ (1/2) - a*b^2*(a^2*c^2)^(1/2) - a^2*b*(a^2*c^2)^(1/2)))/(a*(a + b)^2*(a^2 - b^2)^2*(a - b)^2*(b^2 - a^2 + c^2)))*((a^3*(-(a^2 - b^2)^2*(b^2 - a^2 + c^2))^(1/2))/2 - (b^3*(-(a^2 - b^2)^2*(b^2 - a^2 + c^2))^(1/2))/2 - (a*b^2 *(-(a^2 - b^2)^2*(b^2 - a^2 + c^2))^(1/2))/2 + (a^2*b*(-(a^2 - b^2)^2*(b^2 - a^2 + c^2))^(1/2))/2))*(a^2*c^2)^(1/2))/(-(a^2 - b^2)^2*(b^2 - a^2 + c^ 2))^(1/2)