Integrand size = 17, antiderivative size = 56 \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\frac {\sqrt {2} (A-B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a+a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a+a \cosh (x)}} \] Output:
2^(1/2)*(A-B)*arctan(1/2*a^(1/2)*sinh(x)*2^(1/2)/(a+a*cosh(x))^(1/2))/a^(1 /2)+2*B*sinh(x)/(a+a*cosh(x))^(1/2)
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\frac {2 \cosh \left (\frac {x}{2}\right ) \left ((A-B) \arctan \left (\sinh \left (\frac {x}{2}\right )\right )+2 B \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {a (1+\cosh (x))}} \] Input:
Integrate[(A + B*Cosh[x])/Sqrt[a + a*Cosh[x]],x]
Output:
(2*Cosh[x/2]*((A - B)*ArcTan[Sinh[x/2]] + 2*B*Sinh[x/2]))/Sqrt[a*(1 + Cosh [x])]
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3230, 3042, 3128, 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.
\(\displaystyle \int \frac {A+B \cosh (x)}{\sqrt {a \cosh (x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{\sqrt {a+a \sin \left (\frac {\pi }{2}+i x\right )}}dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle (A-B) \int \frac {1}{\sqrt {\cosh (x) a+a}}dx+\frac {2 B \sinh (x)}{\sqrt {a \cosh (x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 B \sinh (x)}{\sqrt {a \cosh (x)+a}}+(A-B) \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right ) a+a}}dx\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {2 B \sinh (x)}{\sqrt {a \cosh (x)+a}}+2 i (A-B) \int \frac {1}{\frac {a^2 \sinh ^2(x)}{\cosh (x) a+a}+2 a}d\left (-\frac {i a \sinh (x)}{\sqrt {\cosh (x) a+a}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {2} (A-B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a \cosh (x)+a}}\) |
Input:
Int[(A + B*Cosh[x])/Sqrt[a + a*Cosh[x]],x]
Output:
(Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/ Sqrt[a] + (2*B*Sinh[x])/Sqrt[a + a*Cosh[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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(45)=90\).
Time = 0.91 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.29
method | result | size |
default | \(-\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (\ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a A -2 B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-\ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a B \right ) \sqrt {2}}{\sqrt {-a}\, a \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(128\) |
parts | \(-\frac {A \cosh \left (\frac {x}{2}\right ) \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \sqrt {2}}{\sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}+\frac {B \cosh \left (\frac {x}{2}\right ) \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (\ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a +2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{\sqrt {-a}\, a \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(164\) |
Input:
int((A+B*cosh(x))/(a+cosh(x)*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-cosh(1/2*x)*(sinh(1/2*x)^2*a)^(1/2)*(ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^ (1/2)*(-a)^(1/2)-a))*a*A-2*B*(sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)-ln(2/cosh( 1/2*x)*((sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)-a))*a*B)/(-a)^(1/2)/a/sinh(1/2* x)*2^(1/2)/(a*cosh(1/2*x)^2)^(1/2)
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (A - B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{\sqrt {a}}\right ) - \sqrt {\frac {1}{2}} {\left (B \cosh \left (x\right ) + B \sinh \left (x\right ) - B\right )} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}\right )}}{a} \] Input:
integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="fricas")
Output:
-2*(sqrt(2)*(A - B)*sqrt(a)*arctan(sqrt(2)*sqrt(1/2)*sqrt(a/(cosh(x) + sin h(x)))/sqrt(a)) - sqrt(1/2)*(B*cosh(x) + B*sinh(x) - B)*sqrt(a/(cosh(x) + sinh(x))))/a
\[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\int \frac {A + B \cosh {\left (x \right )}}{\sqrt {a \left (\cosh {\left (x \right )} + 1\right )}}\, dx \] Input:
integrate((A+B*cosh(x))/(a+a*cosh(x))**(1/2),x)
Output:
Integral((A + B*cosh(x))/sqrt(a*(cosh(x) + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (45) = 90\).
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.11 \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=2 \, {\left (\sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} + \frac {e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a} e^{x} + \sqrt {a}}\right )} - \frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a} e^{x} + \sqrt {a}}\right )} A - \frac {1}{3} \, {\left (3 \, \sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} - \frac {e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a} e^{x} + \sqrt {a}}\right )} - \sqrt {2} {\left (\frac {3 \, \arctan \left (e^{\left (-\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} - \frac {2 \, e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a}} - \frac {e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a} e^{\left (-x\right )} + \sqrt {a}}\right )} - \frac {3 \, \sqrt {2} \sqrt {a} e^{\left (\frac {3}{2} \, x\right )} - \sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, x\right )}}{a e^{x} + a}\right )} B \] Input:
integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="maxima")
Output:
2*(sqrt(2)*(arctan(e^(1/2*x))/sqrt(a) + e^(1/2*x)/(sqrt(a)*e^x + sqrt(a))) - sqrt(2)*e^(1/2*x)/(sqrt(a)*e^x + sqrt(a)))*A - 1/3*(3*sqrt(2)*(arctan(e ^(1/2*x))/sqrt(a) - e^(1/2*x)/(sqrt(a)*e^x + sqrt(a))) - sqrt(2)*(3*arctan (e^(-1/2*x))/sqrt(a) - 2*e^(-1/2*x)/sqrt(a) - e^(-1/2*x)/(sqrt(a)*e^(-x) + sqrt(a))) - (3*sqrt(2)*sqrt(a)*e^(3/2*x) - sqrt(2)*sqrt(a)*e^(-1/2*x))/(a *e^x + a))*B
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\frac {2 \, \sqrt {2} {\left (A - B\right )} \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} + \frac {\sqrt {2} B e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a}} - \frac {\sqrt {2} B e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a}} \] Input:
integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="giac")
Output:
2*sqrt(2)*(A - B)*arctan(e^(1/2*x))/sqrt(a) + sqrt(2)*B*e^(1/2*x)/sqrt(a) - sqrt(2)*B*e^(-1/2*x)/sqrt(a)
Timed out. \[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{\sqrt {a+a\,\mathrm {cosh}\left (x\right )}} \,d x \] Input:
int((A + B*cosh(x))/(a + a*cosh(x))^(1/2),x)
Output:
int((A + B*cosh(x))/(a + a*cosh(x))^(1/2), x)
\[ \int \frac {A+B \cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\cosh \left (x \right )+1}}{\cosh \left (x \right )+1}d x \right ) a +\left (\int \frac {\sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )}{\cosh \left (x \right )+1}d x \right ) b \right )}{a} \] Input:
int((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x)
Output:
(sqrt(a)*(int(sqrt(cosh(x) + 1)/(cosh(x) + 1),x)*a + int((sqrt(cosh(x) + 1 )*cosh(x))/(cosh(x) + 1),x)*b))/a