Integrand size = 21, antiderivative size = 54 \[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}+\frac {\text {arctanh}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b} \] Output:
-arctan(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2))/b+arctanh(cosh(b*x+a)^(1/2)/s inh(b*x+a)^(1/2))/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\frac {2 \sqrt [4]{\cosh ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},-\sinh ^2(a+b x)\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}} \] Input:
Integrate[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]],x]
Output:
(2*(Cosh[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -Sinh[a + b*x] ^2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[Cosh[a + b*x]])
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3055, 827, 216, 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.
\(\displaystyle \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\cos (i a+i b x)}}{\sqrt {-i \sin (i a+i b x)}}dx\) |
\(\Big \downarrow \) 3055 |
\(\displaystyle \frac {2 \int \frac {\coth (a+b x)}{1-\coth ^2(a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}}{b}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}-\frac {1}{2} \int \frac {1}{\coth (a+b x)+1}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}-\frac {1}{2} \arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )\right )}{b}\) |
Input:
Int[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]],x]
Output:
(2*(-1/2*ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]] + ArcTanh[Sqrt[Co sh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/2))/b
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> With[{k = Denominator[m]}, Simp[(-k)*a*(b/f) Subst[Int[x ^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[ e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
\[\int \frac {\sqrt {\cosh \left (b x +a \right )}}{\sqrt {\sinh \left (b x +a \right )}}d x\]
Input:
int(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x)
Output:
int(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (46) = 92\).
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.67 \[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\frac {2 \, \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right )}{2 \, b} \] Input:
integrate(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x, algorithm="fricas")
Output:
1/2*(2*arctan(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt(co sh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh(b* x + a)^2) - log(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt( cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh( b*x + a)^2))/b
\[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\int \frac {\sqrt {\cosh {\left (a + b x \right )}}}{\sqrt {\sinh {\left (a + b x \right )}}}\, dx \] Input:
integrate(cosh(b*x+a)**(1/2)/sinh(b*x+a)**(1/2),x)
Output:
Integral(sqrt(cosh(a + b*x))/sqrt(sinh(a + b*x)), x)
\[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\int { \frac {\sqrt {\cosh \left (b x + a\right )}}{\sqrt {\sinh \left (b x + a\right )}} \,d x } \] Input:
integrate(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(cosh(b*x + a))/sqrt(sinh(b*x + a)), x)
\[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\int { \frac {\sqrt {\cosh \left (b x + a\right )}}{\sqrt {\sinh \left (b x + a\right )}} \,d x } \] Input:
integrate(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(cosh(b*x + a))/sqrt(sinh(b*x + a)), x)
Timed out. \[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\int \frac {\sqrt {\mathrm {cosh}\left (a+b\,x\right )}}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \] Input:
int(cosh(a + b*x)^(1/2)/sinh(a + b*x)^(1/2),x)
Output:
int(cosh(a + b*x)^(1/2)/sinh(a + b*x)^(1/2), x)
\[ \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx=\int \frac {\sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}}{\sinh \left (b x +a \right )}d x \] Input:
int(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2),x)
Output:
int((sqrt(sinh(a + b*x))*sqrt(cosh(a + b*x)))/sinh(a + b*x),x)