Integrand size = 18, antiderivative size = 80 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\frac {b B x}{b^2-c^2}+\frac {A \arctan \left (\frac {c \cosh (x)+b \sinh (x)}{\sqrt {b^2-c^2}}\right )}{\sqrt {b^2-c^2}}-\frac {B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]
b*B*x/(b^2-c^2)-B*c*ln(b*cosh(x)+c*sinh(x))/(b^2-c^2)+A*arctan((c*cosh(x)+ b*sinh(x))/(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\frac {b B x+2 A \sqrt {b-c} \sqrt {b+c} \arctan \left (\frac {c+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-c} \sqrt {b+c}}\right )-B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]
(b*B*x + 2*A*Sqrt[b - c]*Sqrt[b + c]*ArcTan[(c + b*Tanh[x/2])/(Sqrt[b - c] *Sqrt[b + c])] - B*c*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3042, 3617, 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.
\(\displaystyle \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \cos (i x)}{b \cos (i x)-i c \sin (i x)}dx\) |
\(\Big \downarrow \) 3617 |
\(\displaystyle A \int \frac {1}{b \cosh (x)+c \sinh (x)}dx+\frac {b B x}{b^2-c^2}-\frac {B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle A \int \frac {1}{b \cos (i x)-i c \sin (i x)}dx+\frac {b B x}{b^2-c^2}-\frac {B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle i A \int \frac {1}{b^2-c^2-(-i c \cosh (x)-i b \sinh (x))^2}d(-i c \cosh (x)-i b \sinh (x))+\frac {b B x}{b^2-c^2}-\frac {B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {i A \text {arctanh}\left (\frac {-i b \sinh (x)-i c \cosh (x)}{\sqrt {b^2-c^2}}\right )}{\sqrt {b^2-c^2}}+\frac {b B x}{b^2-c^2}-\frac {B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\) |
(b*B*x)/(b^2 - c^2) + (I*A*ArcTanh[((-I)*c*Cosh[x] - I*b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] - (B*c*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)
3.8.27.3.1 Defintions of rubi rules used
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_.) + 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]
Time = 0.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58
method | result | size |
default | \(-\frac {2 B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b +2 c}+\frac {2 B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b -2 c}+\frac {-B c \ln \left (\tanh \left (\frac {x}{2}\right )^{2} b +2 c \tanh \left (\frac {x}{2}\right )+b \right )+\frac {2 \left (b^{2} A -A \,c^{2}\right ) \arctan \left (\frac {2 b \tanh \left (\frac {x}{2}\right )+2 c}{2 \sqrt {b^{2}-c^{2}}}\right )}{\sqrt {b^{2}-c^{2}}}}{\left (b -c \right ) \left (b +c \right )}\) | \(126\) |
risch | \(\frac {B x}{b +c}+\frac {2 x B \,b^{2} c}{b^{4}-2 b^{2} c^{2}+c^{4}}-\frac {2 x B \,c^{3}}{b^{4}-2 b^{2} c^{2}+c^{4}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{A \left (b +c \right )}\right ) B c}{\left (b +c \right ) \left (b -c \right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{A \left (b +c \right )}\right ) \sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{\left (b +c \right ) \left (b -c \right )}-\frac {\ln \left ({\mathrm e}^{x}-\frac {\sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{A \left (b +c \right )}\right ) B c}{\left (b +c \right ) \left (b -c \right )}-\frac {\ln \left ({\mathrm e}^{x}-\frac {\sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{A \left (b +c \right )}\right ) \sqrt {-A^{2} b^{2}+A^{2} c^{2}}}{\left (b +c \right ) \left (b -c \right )}\) | \(280\) |
-2*B/(2*b+2*c)*ln(tanh(1/2*x)-1)+2*B/(2*b-2*c)*ln(tanh(1/2*x)+1)+2/(b-c)/( b+c)*(-1/2*B*c*ln(tanh(1/2*x)^2*b+2*c*tanh(1/2*x)+b)+(A*b^2-A*c^2)/(b^2-c^ 2)^(1/2)*arctan(1/2*(2*b*tanh(1/2*x)+2*c)/(b^2-c^2)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.92 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\left [-\frac {B c \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \sqrt {-b^{2} + c^{2}} A \log \left (\frac {{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-b^{2} + c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - b + c}{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} + b - c}\right ) - {\left (B b + B c\right )} x}{b^{2} - c^{2}}, -\frac {B c \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, \sqrt {b^{2} - c^{2}} A \arctan \left (\frac {\sqrt {b^{2} - c^{2}}}{{\left (b + c\right )} \cosh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )}\right ) - {\left (B b + B c\right )} x}{b^{2} - c^{2}}\right ] \]
[-(B*c*log(2*(b*cosh(x) + c*sinh(x))/(cosh(x) - sinh(x))) + sqrt(-b^2 + c^ 2)*A*log(((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^ 2 - 2*sqrt(-b^2 + c^2)*(cosh(x) + sinh(x)) - b + c)/((b + c)*cosh(x)^2 + 2 *(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 + b - c)) - (B*b + B*c)*x)/(b ^2 - c^2), -(B*c*log(2*(b*cosh(x) + c*sinh(x))/(cosh(x) - sinh(x))) + 2*sq rt(b^2 - c^2)*A*arctan(sqrt(b^2 - c^2)/((b + c)*cosh(x) + (b + c)*sinh(x)) ) - (B*b + B*c)*x)/(b^2 - c^2)]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (66) = 132\).
Time = 30.01 (sec) , antiderivative size = 697, normalized size of antiderivative = 8.71 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\begin {cases} \tilde {\infty } \left (A \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x - 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {A \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x - 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{c} & \text {for}\: b = 0 \\- \frac {2 A}{- 2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} + \frac {B x \sinh {\left (x \right )}}{- 2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} - \frac {B x \cosh {\left (x \right )}}{- 2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} - \frac {B \cosh {\left (x \right )}}{- 2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} & \text {for}\: b = - c \\- \frac {2 A}{2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} + \frac {B x \sinh {\left (x \right )}}{2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} + \frac {B x \cosh {\left (x \right )}}{2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} - \frac {B \cosh {\left (x \right )}}{2 c \sinh {\left (x \right )} + 2 c \cosh {\left (x \right )}} & \text {for}\: b = c \\\frac {A b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} - \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} - \frac {A b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} + \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} - \frac {A c^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} - \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} + \frac {A c^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} + \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} + \frac {B b x \sqrt {- b^{2} + c^{2}}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} - \frac {B c x \sqrt {- b^{2} + c^{2}}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} + \frac {2 B c \sqrt {- b^{2} + c^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} - \frac {B c \sqrt {- b^{2} + c^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} - \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} - \frac {B c \sqrt {- b^{2} + c^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {c}{b} + \frac {\sqrt {- b^{2} + c^{2}}}{b} \right )}}{b^{2} \sqrt {- b^{2} + c^{2}} - c^{2} \sqrt {- b^{2} + c^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(A*log(tanh(x/2)) + B*x - 2*B*log(tanh(x/2) + 1) + B*log(ta nh(x/2))), Eq(b, 0) & Eq(c, 0)), ((A*log(tanh(x/2)) + B*x - 2*B*log(tanh(x /2) + 1) + B*log(tanh(x/2)))/c, Eq(b, 0)), (-2*A/(-2*c*sinh(x) + 2*c*cosh( x)) + B*x*sinh(x)/(-2*c*sinh(x) + 2*c*cosh(x)) - B*x*cosh(x)/(-2*c*sinh(x) + 2*c*cosh(x)) - B*cosh(x)/(-2*c*sinh(x) + 2*c*cosh(x)), Eq(b, -c)), (-2* A/(2*c*sinh(x) + 2*c*cosh(x)) + B*x*sinh(x)/(2*c*sinh(x) + 2*c*cosh(x)) + B*x*cosh(x)/(2*c*sinh(x) + 2*c*cosh(x)) - B*cosh(x)/(2*c*sinh(x) + 2*c*cos h(x)), Eq(b, c)), (A*b**2*log(tanh(x/2) + c/b - sqrt(-b**2 + c**2)/b)/(b** 2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) - A*b**2*log(tanh(x/2) + c /b + sqrt(-b**2 + c**2)/b)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c* *2)) - A*c**2*log(tanh(x/2) + c/b - sqrt(-b**2 + c**2)/b)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) + A*c**2*log(tanh(x/2) + c/b + sqrt(-b **2 + c**2)/b)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) + B*b*x *sqrt(-b**2 + c**2)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) - B*c*x*sqrt(-b**2 + c**2)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2 )) + 2*B*c*sqrt(-b**2 + c**2)*log(tanh(x/2) + 1)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) - B*c*sqrt(-b**2 + c**2)*log(tanh(x/2) + c/b - sqrt(-b**2 + c**2)/b)/(b**2*sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)) - B*c*sqrt(-b**2 + c**2)*log(tanh(x/2) + c/b + sqrt(-b**2 + c**2)/b)/(b**2 *sqrt(-b**2 + c**2) - c**2*sqrt(-b**2 + c**2)), True))
Exception generated. \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\text {Exception raised: ValueError} \]
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*c^2-4*b^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=-\frac {B c \log \left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}{b^{2} - c^{2}} + \frac {2 \, A \arctan \left (\frac {b e^{x} + c e^{x}}{\sqrt {b^{2} - c^{2}}}\right )}{\sqrt {b^{2} - c^{2}}} + \frac {B x}{b - c} \]
-B*c*log(b*e^(2*x) + c*e^(2*x) + b - c)/(b^2 - c^2) + 2*A*arctan((b*e^x + c*e^x)/sqrt(b^2 - c^2))/sqrt(b^2 - c^2) + B*x/(b - c)
Time = 3.89 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.21 \[ \int \frac {A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {A\,{\mathrm {e}}^x\,\sqrt {b^2-c^2}}{b\,\sqrt {A^2}-c\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {b^2-c^2}}+\frac {B\,x}{b-c}+\frac {B\,c^3\,\ln \left (4\,A^2\,b-4\,A^2\,c+4\,A^2\,b\,{\mathrm {e}}^{2\,x}+4\,A^2\,c\,{\mathrm {e}}^{2\,x}\right )}{b^4-2\,b^2\,c^2+c^4}-\frac {B\,b^2\,c\,\ln \left (4\,A^2\,b-4\,A^2\,c+4\,A^2\,b\,{\mathrm {e}}^{2\,x}+4\,A^2\,c\,{\mathrm {e}}^{2\,x}\right )}{b^4-2\,b^2\,c^2+c^4} \]
(2*atan((A*exp(x)*(b^2 - c^2)^(1/2))/(b*(A^2)^(1/2) - c*(A^2)^(1/2)))*(A^2 )^(1/2))/(b^2 - c^2)^(1/2) + (B*x)/(b - c) + (B*c^3*log(4*A^2*b - 4*A^2*c + 4*A^2*b*exp(2*x) + 4*A^2*c*exp(2*x)))/(b^4 + c^4 - 2*b^2*c^2) - (B*b^2*c *log(4*A^2*b - 4*A^2*c + 4*A^2*b*exp(2*x) + 4*A^2*c*exp(2*x)))/(b^4 + c^4 - 2*b^2*c^2)