3.8.0 ∫ ∘ ____________________ −√ _____ b2 − c2 + b cosh(x) + c sinh(x) dx [776]

Integrand size = 28, antiderivative size = 39 \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 (c \cosh (x)+b \sinh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

Rubi [A] (veriﬁed)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3191} \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 (b \sinh (x)+c \cosh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (c \cosh (x)+b \sinh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 35.97 (sec) , antiderivative size = 4196, normalized size of antiderivative = 107.59 \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\text {Result too large to show} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(35)=70\).

Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.36


method	result	size

risch	\(-\frac {\sqrt {2}\, \sqrt {-\left (-{\mathrm e}^{2 x} b -{\mathrm e}^{2 x} c +2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}-b +c \right ) {\mathrm e}^{-x}}\, \left ({\mathrm e}^{x} b +{\mathrm e}^{x} c +\sqrt {b^{2}-c^{2}}\right ) \left ({\mathrm e}^{x} b +{\mathrm e}^{x} c -\sqrt {b^{2}-c^{2}}\right )}{\left (-{\mathrm e}^{2 x} b -{\mathrm e}^{2 x} c +2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}-b +c \right ) \left (b +c \right )}\)	\(131\)
default	\(\frac {\left (-b^{2}+c^{2}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(202\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (35) = 70\).

Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.67 \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} - b + c} \]

Sympy [F]

\[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\int \sqrt {b \cosh {\left (x \right )} + c \sinh {\left (x \right )} - \sqrt {b^{2} - c^{2}}}\, dx \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (35) = 70\).

Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.00 \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=-\frac {\sqrt {2} \sqrt {-2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} \sqrt {b + c} \sqrt {b - c} e^{\left (\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt {b + c} \sqrt {b - c}} - \frac {\sqrt {2} \sqrt {-2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} {\left (b - c\right )} e^{\left (-\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt {b + c} \sqrt {b - c}} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (35) = 70\).

Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.08 \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=-\frac {\sqrt {2} {\left ({\left (b - c\right )} e^{\left (-\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + \sqrt {b^{2} - c^{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )}}{\sqrt {b - c}} \]

Mupad [F(-1)]

Timed out. \[ \int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx=\int \sqrt {b\,\mathrm {cosh}\left (x\right )-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )} \,d x \]

\(\int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\) [776]

Optimal result