Optimal. Leaf size=82 \[ -\frac{-A b \sinh (x)-A c \cosh (x)+b C}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}-\frac{c C \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}} \]
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Rubi [A] time = 0.0790252, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3154, 3074, 206} \[ -\frac{-A b \sinh (x)-A c \cosh (x)+b C}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}-\frac{c C \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3154
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{A+C \sinh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx &=-\frac{b C-A c \cosh (x)-A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}-\frac{(c C) \int \frac{1}{b \cosh (x)+c \sinh (x)} \, dx}{b^2-c^2}\\ &=-\frac{b C-A c \cosh (x)-A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}-\frac{(i c C) \operatorname{Subst}\left (\int \frac{1}{b^2-c^2-x^2} \, dx,x,-i c \cosh (x)-i b \sinh (x)\right )}{b^2-c^2}\\ &=-\frac{c C \tan ^{-1}\left (\frac{c \cosh (x)+b \sinh (x)}{\sqrt{b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}}-\frac{b C-A c \cosh (x)-A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.439, size = 155, normalized size = 1.89 \[ -\frac{\sinh (x) \left (2 b c^2 C \sqrt{b+c} \tan ^{-1}\left (\frac{b \tanh \left (\frac{x}{2}\right )+c}{\sqrt{b-c} \sqrt{b+c}}\right )-A (b-c)^{3/2} (b+c)^2\right )+2 b^2 c C \sqrt{b+c} \cosh (x) \tan ^{-1}\left (\frac{b \tanh \left (\frac{x}{2}\right )+c}{\sqrt{b-c} \sqrt{b+c}}\right )+b^2 C \sqrt{b-c} (b+c)}{b (b-c)^{3/2} (b+c)^2 (b \cosh (x)+c \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 115, normalized size = 1.4 \begin{align*} -2\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+2\,c\tanh \left ( x/2 \right ) +b} \left ( -{\frac{ \left ( A{b}^{2}-A{c}^{2}-Ccb \right ) \tanh \left ( x/2 \right ) }{ \left ({b}^{2}-{c}^{2} \right ) b}}+{\frac{bC}{{b}^{2}-{c}^{2}}} \right ) }-2\,{\frac{Cc}{ \left ({b}^{2}-{c}^{2} \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b+2\,c}{\sqrt{{b}^{2}-{c}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42513, size = 1648, normalized size = 20.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14731, size = 112, normalized size = 1.37 \begin{align*} -\frac{2 \, C c \arctan \left (\frac{b e^{x} + c e^{x}}{\sqrt{b^{2} - c^{2}}}\right )}{{\left (b^{2} - c^{2}\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (C b e^{x} + A b - A c\right )}}{{\left (b^{2} - c^{2}\right )}{\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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