Optimal. Leaf size=28 \[ \frac{(b c-a d) \log (c+d \coth (x))}{d^2}-\frac{b \coth (x)}{d} \]
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Rubi [A] time = 0.0967657, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4344, 43} \[ \frac{(b c-a d) \log (c+d \coth (x))}{d^2}-\frac{b \coth (x)}{d} \]
Antiderivative was successfully verified.
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Rule 4344
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b \coth (x)) \text{csch}^2(x)}{c+d \coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b x}{c+d x} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac{b \coth (x)}{d}+\frac{(b c-a d) \log (c+d \coth (x))}{d^2}\\ \end{align*}
Mathematica [A] time = 0.300637, size = 56, normalized size = 2. \[ \frac{\sinh (x) (a+b \coth (x)) (-(b c-a d) (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-b d \coth (x))}{d^2 (a \sinh (x)+b \cosh (x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 94, normalized size = 3.4 \begin{align*} -{\frac{b}{2\,d}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{a}{d}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }+{\frac{cb}{{d}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }-{\frac{b}{2\,d} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{a}{d}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{cb}{{d}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19942, size = 104, normalized size = 3.71 \begin{align*} b{\left (\frac{c \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{2}} - \frac{c \log \left (e^{\left (-x\right )} + 1\right )}{d^{2}} - \frac{c \log \left (e^{\left (-x\right )} - 1\right )}{d^{2}} + \frac{2}{d e^{\left (-2 \, x\right )} - d}\right )} - \frac{a \log \left (d \coth \left (x\right ) + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20974, size = 467, normalized size = 16.68 \begin{align*} -\frac{2 \, b d -{\left ({\left (b c - a d\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b c - a d\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b c - a d\right )} \sinh \left (x\right )^{2} - b c + a d\right )} \log \left (\frac{2 \,{\left (d \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left ({\left (b c - a d\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b c - a d\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b c - a d\right )} \sinh \left (x\right )^{2} - b c + a d\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{d^{2} \cosh \left (x\right )^{2} + 2 \, d^{2} \cosh \left (x\right ) \sinh \left (x\right ) + d^{2} \sinh \left (x\right )^{2} - d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \coth{\left (x \right )}\right ) \operatorname{csch}^{2}{\left (x \right )}}{c + d \coth{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12092, size = 153, normalized size = 5.46 \begin{align*} \frac{{\left (b c^{2} - a c d + b c d - a d^{2}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{2} + d^{3}} - \frac{{\left (b c - a d\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{2}} + \frac{b c e^{\left (2 \, x\right )} - a d e^{\left (2 \, x\right )} - b c + a d - 2 \, b d}{d^{2}{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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