Optimal. Leaf size=96 \[ -\frac {i \sqrt {2} F\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{d \sqrt {2 a+b \sinh (2 c+2 d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2745, 2742,
2740} \begin {gather*} -\frac {i \sqrt {2} \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {2 a+b \sinh (2 c+2 d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2740
Rule 2742
Rule 2745
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \cosh (c+d x) \sinh (c+d x)}} \, dx &=\int \frac {1}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac {\sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}} \int \frac {1}{\sqrt {\frac {a}{a-\frac {i b}{2}}+\frac {b \sinh (2 c+2 d x)}{2 \left (a-\frac {i b}{2}\right )}}} \, dx}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}}\\ &=-\frac {i \sqrt {2} F\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{d \sqrt {2 a+b \sinh (2 c+2 d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 90, normalized size = 0.94 \begin {gather*} \frac {i F\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}}{d \sqrt {a+\frac {1}{2} b \sinh (2 (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.20, size = 181, normalized size = 1.89
method | result | size |
default | \(-\frac {2 \left (i b -2 a \right ) \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right )}{b \cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.08, size = 142, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (\sqrt {-b} b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a \sqrt {-b}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} {\rm ellipticF}\left (\sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}, \frac {4 \, a b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 8 \, a^{2} - b^{2}}{b^{2}}\right )}{b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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