Optimal. Leaf size=22 \[ \frac {c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {3229}
\begin {gather*} \frac {b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3229
Rubi steps
\begin {align*} \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx &=\frac {c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 34, normalized size = 1.55 \begin {gather*} \frac {-a c+b^2 \sinh (x)-c^2 \sinh (x)}{b (a+b \cosh (x)+c \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs.
\(2(22)=44\).
time = 1.42, size = 73, normalized size = 3.32
method | result | size |
risch | \(-\frac {2 \left (a \,{\mathrm e}^{x}+b -c \right )}{b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +2 a \,{\mathrm e}^{x}+b -c}\) | \(36\) |
default | \(\frac {-\frac {2 \left (a b -b^{2}+c^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a -b}-\frac {2 a c}{a -b}}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 c \tanh \left (\frac {x}{2}\right )-a -b}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 55 vs.
\(2 (22) = 44\).
time = 0.38, size = 55, normalized size = 2.50 \begin {gather*} -\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b - c\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 35, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (a e^{x} + b - c\right )}}{b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c} \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 = -0.05 \begin {gather*} \int \frac {b^2+a\,\mathrm {cosh}\left (x\right )\,b-c^2+a\,\mathrm {sinh}\left (x\right )\,c}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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