Optimal. Leaf size=54 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2655, 304, 209,
212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}-\frac {\text {ArcTan}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 2655
Rubi steps
\begin {align*} \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 57, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt [4]{\cosh ^2(a+b x)} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\sinh ^2(a+b x)\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.68, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\cosh }\left (b x +a \right )}{\sqrt {\sinh \left (b x +a \right )}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (46) = 92\).
time = 0.38, size = 144, normalized size = 2.67 \begin {gather*} \frac {2 \, \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right )}{2 \, b} \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 {\sqrt {\cosh {\left (a + b x \right )}}}{\sqrt {\sinh {\left (a + b x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\mathrm {cosh}\left (a+b\,x\right )}}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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