Optimal. Leaf size=46 \[ \frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{\sqrt {a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2728, 212}
\begin {gather*} \frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{\sqrt {a} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2728
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \cosh (c+d x)}} \, dx &=\frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {i a \sinh (c+d x)}{\sqrt {a+a \cosh (c+d x)}}\right )}{d}\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{\sqrt {a} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 40, normalized size = 0.87 \begin {gather*} \frac {2 \text {ArcTan}\left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {a (1+\cosh (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs.
\(2(37)=74\).
time = 0.94, size = 103, normalized size = 2.24
method | result | size |
default | \(-\frac {\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\, \ln \left (\frac {2 \sqrt {-a}\, \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) \sqrt {2}}{\sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (37) = 74\).
time = 0.53, size = 86, normalized size = 1.87 \begin {gather*} 2 \, \sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d} + \frac {e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a}\right )} d}\right )} - \frac {2 \, \sqrt {2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d e^{\left (d x + c\right )} + \sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.42, size = 149, normalized size = 3.24 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + \cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{d}, \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{\sqrt {a}}\right )}{\sqrt {a} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \cosh {\left (c + d x \right )} + a}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 21, normalized size = 0.46 \begin {gather*} \frac {2 \, \sqrt {2} \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________