Integrand size = 18, antiderivative size = 78 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\frac {b B \arctan \left (\frac {c \cosh (x)+b \sinh (x)}{\sqrt {b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}}+\frac {B c+A c \cosh (x)+A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))} \] Output:
b*B*arctan((c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2))/(b^2-c^2)^(3/2)+(B*c+A*c *cosh(x)+A*b*sinh(x))/(b^2-c^2)/(b*cosh(x)+c*sinh(x))
Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.94 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\frac {b B \sqrt {b-c} c (b+c)+2 b^3 B \sqrt {b+c} \arctan \left (\frac {c+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-c} \sqrt {b+c}}\right ) \cosh (x)+\left (A (b-c)^{3/2} (b+c)^2+2 b^2 B c \sqrt {b+c} \arctan \left (\frac {c+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-c} \sqrt {b+c}}\right )\right ) \sinh (x)}{b (b-c)^{3/2} (b+c)^2 (b \cosh (x)+c \sinh (x))} \] Input:
Integrate[(A + B*Cosh[x])/(b*Cosh[x] + c*Sinh[x])^2,x]
Output:
(b*B*Sqrt[b - c]*c*(b + c) + 2*b^3*B*Sqrt[b + c]*ArcTan[(c + b*Tanh[x/2])/ (Sqrt[b - c]*Sqrt[b + c])]*Cosh[x] + (A*(b - c)^(3/2)*(b + c)^2 + 2*b^2*B* c*Sqrt[b + c]*ArcTan[(c + b*Tanh[x/2])/(Sqrt[b - c]*Sqrt[b + c])])*Sinh[x] )/(b*(b - c)^(3/2)*(b + c)^2*(b*Cosh[x] + c*Sinh[x]))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3042, 3634, 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.
\(\displaystyle \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \cos (i x)}{(b \cos (i x)-i c \sin (i x))^2}dx\) |
\(\Big \downarrow \) 3634 |
\(\displaystyle \frac {b B \int \frac {1}{b \cosh (x)+c \sinh (x)}dx}{b^2-c^2}+\frac {A b \sinh (x)+A c \cosh (x)+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A b \sinh (x)+A c \cosh (x)+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac {b B \int \frac {1}{b \cos (i x)-i c \sin (i x)}dx}{b^2-c^2}\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle \frac {A b \sinh (x)+A c \cosh (x)+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac {i b B \int \frac {1}{b^2-c^2-(-i c \cosh (x)-i b \sinh (x))^2}d(-i c \cosh (x)-i b \sinh (x))}{b^2-c^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {A b \sinh (x)+A c \cosh (x)+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac {i b B \text {arctanh}\left (\frac {-i b \sinh (x)-i c \cosh (x)}{\sqrt {b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}}\) |
Input:
Int[(A + B*Cosh[x])/(b*Cosh[x] + c*Sinh[x])^2,x]
Output:
(I*b*B*ArcTanh[((-I)*c*Cosh[x] - I*b*Sinh[x])/Sqrt[b^2 - c^2]])/(b^2 - c^2 )^(3/2) + (B*c + A*c*Cosh[x] + A*b*Sinh[x])/((b^2 - c^2)*(b*Cosh[x] + c*Si nh[x]))
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_)])^2, x_Symbol] :> Simp[(c*B + c*A*Co s[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Simp[(a*A - b*B)/(a^2 - 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[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B, 0]
Time = 1.62 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49
method | result | size |
default | \(-\frac {2 \left (-\frac {\left (A \,b^{2}-A \,c^{2}+B \,c^{2}\right ) \tanh \left (\frac {x}{2}\right )}{b \left (b^{2}-c^{2}\right )}-\frac {B c}{b^{2}-c^{2}}\right )}{b \tanh \left (\frac {x}{2}\right )^{2}+2 c \tanh \left (\frac {x}{2}\right )+b}+\frac {2 b B \arctan \left (\frac {2 b \tanh \left (\frac {x}{2}\right )+2 c}{2 \sqrt {b^{2}-c^{2}}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}\) | \(116\) |
risch | \(-\frac {2 \left (-B c \,{\mathrm e}^{x}+A b -c A \right )}{\left (b -c \right ) \left (b +c \right ) \left (b \,{\mathrm e}^{2 x}+c \,{\mathrm e}^{2 x}+b -c \right )}-\frac {b B \ln \left ({\mathrm e}^{x}-\frac {b -c}{\sqrt {-b^{2}+c^{2}}}\right )}{\sqrt {-b^{2}+c^{2}}\, \left (b +c \right ) \left (b -c \right )}+\frac {b B \ln \left ({\mathrm e}^{x}+\frac {b -c}{\sqrt {-b^{2}+c^{2}}}\right )}{\sqrt {-b^{2}+c^{2}}\, \left (b +c \right ) \left (b -c \right )}\) | \(145\) |
Input:
int((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-2*(-(A*b^2-A*c^2+B*c^2)/b/(b^2-c^2)*tanh(1/2*x)-B*c/(b^2-c^2))/(b*tanh(1/ 2*x)^2+2*c*tanh(1/2*x)+b)+2*b*B/(b^2-c^2)^(3/2)*arctan(1/2*(2*b*tanh(1/2*x )+2*c)/(b^2-c^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (74) = 148\).
Time = 0.14 (sec) , antiderivative size = 680, normalized size of antiderivative = 8.72 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
integrate((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))^2,x, algorithm="fricas")
Output:
[-(2*A*b^3 - 2*A*b^2*c - 2*A*b*c^2 + 2*A*c^3 - (B*b^2 - B*b*c + (B*b^2 + B *b*c)*cosh(x)^2 + 2*(B*b^2 + B*b*c)*cosh(x)*sinh(x) + (B*b^2 + B*b*c)*sinh (x)^2)*sqrt(-b^2 + c^2)*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)) - 2*(B*b^2*c - B*c^3)*cosh(x) - 2*(B*b^2*c - B*c^3)*sinh(x))/(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5 + (b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c ^3 + b*c^4 + c^5)*cosh(x)^2 + 2*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c ^4 + c^5)*cosh(x)*sinh(x) + (b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*sinh(x)^2), -2*(A*b^3 - A*b^2*c - A*b*c^2 + A*c^3 + (B*b^2 - B*b*c + (B*b^2 + B*b*c)*cosh(x)^2 + 2*(B*b^2 + B*b*c)*cosh(x)*sinh(x) + (B*b^2 + B*b*c)*sinh(x)^2)*sqrt(b^2 - c^2)*arctan(sqrt(b^2 - c^2)/((b + c)*cosh(x) + (b + c)*sinh(x))) - (B*b^2*c - B*c^3)*cosh(x) - (B*b^2*c - B*c^3)*sinh(x ))/(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5 + (b^5 + b^4*c - 2*b ^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^2 + 2*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)*sinh(x) + (b^5 + b^4*c - 2*b^3*c^2 - 2*b ^2*c^3 + b*c^4 + c^5)*sinh(x)^2)]
Timed out. \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\text {Timed out} \] Input:
integrate((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))^2,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*c^2-4*b^2>0)', see `assume?` f or more de
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\frac {2 \, B b \arctan \left (\frac {b e^{x} + c e^{x}}{\sqrt {b^{2} - c^{2}}}\right )}{{\left (b^{2} - c^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (B c e^{x} - A b + A c\right )}}{{\left (b^{2} - c^{2}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}} \] Input:
integrate((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))^2,x, algorithm="giac")
Output:
2*B*b*arctan((b*e^x + c*e^x)/sqrt(b^2 - c^2))/(b^2 - c^2)^(3/2) + 2*(B*c*e ^x - A*b + A*c)/((b^2 - c^2)*(b*e^(2*x) + c*e^(2*x) + b - c))
Time = 1.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.15 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx=\frac {B\,b\,\ln \left (\frac {2\,B\,b}{{\left (b+c\right )}^{5/2}\,\sqrt {c-b}}+\frac {2\,B\,b\,{\mathrm {e}}^x}{-b^3-b^2\,c+b\,c^2+c^3}\right )}{{\left (b+c\right )}^{3/2}\,{\left (c-b\right )}^{3/2}}-\frac {B\,b\,\ln \left (\frac {2\,B\,b\,{\mathrm {e}}^x}{-b^3-b^2\,c+b\,c^2+c^3}-\frac {2\,B\,b}{{\left (b+c\right )}^{5/2}\,\sqrt {c-b}}\right )}{{\left (b+c\right )}^{3/2}\,{\left (c-b\right )}^{3/2}}-\frac {\frac {2\,A}{b+c}-\frac {2\,B\,c\,{\mathrm {e}}^x}{\left (b+c\right )\,\left (b-c\right )}}{b-c+{\mathrm {e}}^{2\,x}\,\left (b+c\right )} \] Input:
int((A + B*cosh(x))/(b*cosh(x) + c*sinh(x))^2,x)
Output:
(B*b*log((2*B*b)/((b + c)^(5/2)*(c - b)^(1/2)) + (2*B*b*exp(x))/(b*c^2 - b ^2*c - b^3 + c^3)))/((b + c)^(3/2)*(c - b)^(3/2)) - (B*b*log((2*B*b*exp(x) )/(b*c^2 - b^2*c - b^3 + c^3) - (2*B*b)/((b + c)^(5/2)*(c - b)^(1/2))))/(( b + c)^(3/2)*(c - b)^(3/2)) - ((2*A)/(b + c) - (2*B*c*exp(x))/((b + c)*(b - c)))/(b - c + exp(2*x)*(b + c))
Time = 0.20 (sec) , antiderivative size = 920, normalized size of antiderivative = 11.79 \[ \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int((A+B*cosh(x))/(b*cosh(x)+c*sinh(x))^2,x)
Output:
(2*e**(2*x)*sqrt(b**2 - c**2)*atan((e**x*b + e**x*c)/sqrt(b**2 - c**2))*co sh(x)*b**4*c + 2*e**(2*x)*sqrt(b**2 - c**2)*atan((e**x*b + e**x*c)/sqrt(b* *2 - c**2))*cosh(x)*b**3*c**2 + 2*sqrt(b**2 - c**2)*atan((e**x*b + e**x*c) /sqrt(b**2 - c**2))*cosh(x)*b**4*c - 2*sqrt(b**2 - c**2)*atan((e**x*b + e* *x*c)/sqrt(b**2 - c**2))*cosh(x)*b**3*c**2 + 2*e**(2*x)*sqrt(b**2 - c**2)* atan((e**x*b + e**x*c)/sqrt(b**2 - c**2))*sinh(x)*b**3*c**2 + 2*e**(2*x)*s qrt(b**2 - c**2)*atan((e**x*b + e**x*c)/sqrt(b**2 - c**2))*sinh(x)*b**2*c* *3 + 2*sqrt(b**2 - c**2)*atan((e**x*b + e**x*c)/sqrt(b**2 - c**2))*sinh(x) *b**3*c**2 - 2*sqrt(b**2 - c**2)*atan((e**x*b + e**x*c)/sqrt(b**2 - c**2)) *sinh(x)*b**2*c**3 + 2*e**(2*x)*cosh(x)*a*b**4*c + 2*e**(2*x)*cosh(x)*a*b* *3*c**2 - 2*e**(2*x)*cosh(x)*a*b**2*c**3 - 2*e**(2*x)*cosh(x)*a*b*c**4 + 2 *e**x*cosh(x)*b**6 - 2*e**x*cosh(x)*b**4*c**2 + 2*e**(2*x)*sinh(x)*a*b**3* c**2 + 2*e**(2*x)*sinh(x)*a*b**2*c**3 - 2*e**(2*x)*sinh(x)*a*b*c**4 - 2*e* *(2*x)*sinh(x)*a*c**5 - e**(2*x)*b**6 - e**(2*x)*b**5*c + 2*e**(2*x)*b**4* c**2 + 2*e**(2*x)*b**3*c**3 - e**(2*x)*b**2*c**4 - e**(2*x)*b*c**5 + 2*e** x*sinh(x)*b**5*c - 2*e**x*sinh(x)*b**3*c**3 - b**6 + b**5*c + 2*b**4*c**2 - 2*b**3*c**3 - b**2*c**4 + b*c**5)/(c*(e**(2*x)*cosh(x)*b**6 + e**(2*x)*c osh(x)*b**5*c - 2*e**(2*x)*cosh(x)*b**4*c**2 - 2*e**(2*x)*cosh(x)*b**3*c** 3 + e**(2*x)*cosh(x)*b**2*c**4 + e**(2*x)*cosh(x)*b*c**5 + cosh(x)*b**6 - cosh(x)*b**5*c - 2*cosh(x)*b**4*c**2 + 2*cosh(x)*b**3*c**3 + cosh(x)*b*...