Optimal. Leaf size=39 \[ \frac {2 (b \sinh (x)+c \cosh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3112} \[ \frac {2 (b \sinh (x)+c \cosh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3112
Rubi steps
\begin {align*} \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx &=\frac {2 (c \cosh (x)+b \sinh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 74.76, size = 9771, normalized size = 250.54 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 143, normalized size = 3.67 \[ \frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c}{\cosh \relax (x) + \sinh \relax (x)}}}{{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} - b + c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 103, normalized size = 2.64 \[ -\frac {\sqrt {2} {\left (\sqrt {b^{2} - c^{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + {\left (b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )}}{\sqrt {b - c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.89, size = 202, normalized size = 5.18 \[ \frac {\left (-b^{2}+c^{2}\right ) \cosh \relax (x )}{\sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}}\right ) \sqrt {b^{2}-c^{2}}}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.83, size = 156, normalized size = 4.00 \[ -\frac {\sqrt {2} \sqrt {-2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} \sqrt {b + c} \sqrt {b - c} e^{\left (\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt {b + c} \sqrt {b - c}} - \frac {\sqrt {2} \sqrt {-2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} {\left (b - c\right )} e^{\left (-\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt {b + c} \sqrt {b - c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {b\,\mathrm {cosh}\relax (x)-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh {\relax (x )} + c \sinh {\relax (x )} - \sqrt {b^{2} - c^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________