Optimal. Leaf size=71 \[ \frac {B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {\sinh (x) (b B-c C)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3156, 12, 3075} \[ \frac {B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {\sinh (x) (b B-c C)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3075
Rule 3156
Rubi steps
\begin {align*} \int \frac {B \cosh (x)+C \sinh (x)}{(b \cosh (x)+c \sinh (x))^3} \, dx &=\frac {B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 (b B-c C)}{(b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (b^2-c^2\right )}\\ &=\frac {B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {(b B-c C) \int \frac {1}{(b \cosh (x)+c \sinh (x))^2} \, dx}{b^2-c^2}\\ &=\frac {B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {(b B-c C) \sinh (x)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 70, normalized size = 0.99 \[ \frac {C \left (c^2-b^2\right )+b \sinh (2 x) (b B-c C)+c \cosh (2 x) (b B-c C)}{2 b (b-c) (b+c) (b \cosh (x)+c \sinh (x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 232, normalized size = 3.27 \[ -\frac {2 \, {\left ({\left ({\left (2 \, B + C\right )} b + B c\right )} \cosh \relax (x) + {\left (C b + {\left (B + 2 \, C\right )} c\right )} \sinh \relax (x)\right )}}{{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{3} + 3 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \relax (x)^{3} + {\left (3 \, b^{4} + 4 \, b^{3} c - 2 \, b^{2} c^{2} - 4 \, b c^{3} - c^{4}\right )} \cosh \relax (x) + {\left (b^{4} + 4 \, b^{3} c + 2 \, b^{2} c^{2} - 4 \, b c^{3} - 3 \, c^{4} + 3 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 70, normalized size = 0.99 \[ -\frac {2 \, {\left (B b e^{\left (2 \, x\right )} + C b e^{\left (2 \, x\right )} + B c e^{\left (2 \, x\right )} + C c e^{\left (2 \, x\right )} + B b - C c\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 63, normalized size = 0.89 \[ -\frac {2 \left (-\frac {B \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{b}-\frac {\left (B c +b C \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{b^{2}}-\frac {B \tanh \left (\frac {x}{2}\right )}{b}\right )}{\left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 c \tanh \left (\frac {x}{2}\right )+b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 337, normalized size = 4.75 \[ 2 \, B {\left (\frac {{\left (b - c\right )} e^{\left (-2 \, x\right )}}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} + {\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}} + \frac {b}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} + {\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}}\right )} - 2 \, C {\left (\frac {{\left (b - c\right )} e^{\left (-2 \, x\right )}}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} + {\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}} + \frac {c}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} + {\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.66, size = 67, normalized size = 0.94 \[ -\frac {b\,\left (2\,B+2\,B\,{\mathrm {e}}^{2\,x}+2\,C\,{\mathrm {e}}^{2\,x}\right )+c\,\left (2\,B\,{\mathrm {e}}^{2\,x}-2\,C+2\,C\,{\mathrm {e}}^{2\,x}\right )}{{\left (b+c\right )}^2\,{\left (b-c+b\,{\mathrm {e}}^{2\,x}+c\,{\mathrm {e}}^{2\,x}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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