Optimal. Leaf size=100 \[ -\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2831, 2742,
2740, 2734, 2732} \begin {gather*} \frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx &=\frac {\int \sqrt {a+b \cosh (x)} \, dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{b}\\ &=\frac {\sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{b \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 73, normalized size = 0.73 \begin {gather*} -\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \left ((a+b) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-a F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{b \sqrt {a+b \cosh (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.34, size = 181, normalized size = 1.81
method | result | size |
default | \(\frac {2 \left (\EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )\right ) \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) b +\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}}\) | \(181\) |
risch | \(\frac {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) \sqrt {2}\, {\mathrm e}^{-x}}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{-x}}}+\frac {\left (-\frac {4 \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right )}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{x}}}+\frac {4 \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {{\mathrm e}^{x} b}{a +\sqrt {a^{2}-b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) \EllipticE \left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}-b^{2}}\right ) \EllipticF \left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {{\mathrm e}^{3 x} b +2 \,{\mathrm e}^{2 x} a +{\mathrm e}^{x} b}}\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{x}}\, {\mathrm e}^{-x}}{2 \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{-x}}}\) | \(558\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.14, size = 174, normalized size = 1.74 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 3 \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {b \cosh \left (x\right ) + a} b\right )}}{3 \, b^{2}} \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 {\cosh {\left (x \right )}}{\sqrt {a + b \cosh {\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {cosh}\left (x\right )}{\sqrt {a+b\,\mathrm {cosh}\left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________