Optimal. Leaf size=88 \[ \frac{A b \sinh (x)+A c \cosh (x)-b C+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac{(b B-c C) \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}} \]
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Rubi [A] time = 0.0686693, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3153, 3074, 206} \[ \frac{A b \sinh (x)+A c \cosh (x)-b C+B c}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac{(b B-c C) \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\left (b^2-c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3153
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)+C \sinh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx &=\frac{B c-b C+A c \cosh (x)+A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac{(b B-c C) \int \frac{1}{b \cosh (x)+c \sinh (x)} \, dx}{b^2-c^2}\\ &=\frac{B c-b C+A c \cosh (x)+A b \sinh (x)}{\left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}+\frac{(i (b B-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{(b B-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-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 = 0.261468, size = 106, normalized size = 1.2 \[ \frac{A \left (b^2-c^2\right ) \sinh (x)+b (B c-b C)}{b (b-c) (b+c) (b \cosh (x)+c \sinh (x))}+\frac{2 (b B-c C) \tan ^{-1}\left (\frac{b \tanh \left (\frac{x}{2}\right )+c}{\sqrt{b-c} \sqrt{b+c}}\right )}{(b-c)^{3/2} (b+c)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 167, normalized size = 1.9 \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}+B{c}^{2}-Ccb \right ) \tanh \left ( x/2 \right ) }{b \left ({b}^{2}-{c}^{2} \right ) }}-{\frac{Bc-bC}{{b}^{2}-{c}^{2}}} \right ) }+2\,{\frac{Bb}{ \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 ) }-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 = 2.20108, size = 1897, normalized size = 21.56 \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 = 1.20307, size = 128, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (B b - C c\right )} \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} - B c 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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