Optimal. Leaf size=53 \[ \frac{b \coth (x) (b c-a d)}{d^2}-\frac{(b c-a d)^2 \log (c+d \coth (x))}{d^3}-\frac{(a+b \coth (x))^2}{2 d} \]
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Rubi [A] time = 0.145078, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4344, 43} \[ \frac{b \coth (x) (b c-a d)}{d^2}-\frac{(b c-a d)^2 \log (c+d \coth (x))}{d^3}-\frac{(a+b \coth (x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 4344
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b \coth (x))^2 \text{csch}^2(x)}{c+d \coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^2}{c+d x} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx,x,\coth (x)\right )\\ &=\frac{b (b c-a d) \coth (x)}{d^2}-\frac{(a+b \coth (x))^2}{2 d}-\frac{(b c-a d)^2 \log (c+d \coth (x))}{d^3}\\ \end{align*}
Mathematica [A] time = 0.465753, size = 62, normalized size = 1.17 \[ \frac{2 b d \coth (x) (b c-2 a d)+2 (b c-a d)^2 (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-b^2 d^2 \text{csch}^2(x)}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 203, normalized size = 3.8 \begin{align*} -{\frac{{b}^{2}}{8\,d} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{ab}{d}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{c{b}^{2}}{2\,{d}^{2}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{{a}^{2}}{d}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }+2\,{\frac{\ln \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) cba}{{d}^{2}}}-{\frac{{c}^{2}{b}^{2}}{{d}^{3}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }-{\frac{{b}^{2}}{8\,d} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{2}}{d}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-2\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) \right ) cba}{{d}^{2}}}+{\frac{{c}^{2}{b}^{2}}{{d}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{ab}{d} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{c{b}^{2}}{2\,{d}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22232, size = 239, normalized size = 4.51 \begin{align*} b^{2}{\left (\frac{2 \,{\left ({\left (c + d\right )} e^{\left (-2 \, x\right )} - c\right )}}{2 \, d^{2} e^{\left (-2 \, x\right )} - d^{2} e^{\left (-4 \, x\right )} - d^{2}} - \frac{c^{2} \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{3}} + \frac{c^{2} \log \left (e^{\left (-x\right )} + 1\right )}{d^{3}} + \frac{c^{2} \log \left (e^{\left (-x\right )} - 1\right )}{d^{3}}\right )} + 2 \, a 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^{2} \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 = 2.30032, size = 1669, normalized size = 31.49 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \coth{\left (x \right )}\right )^{2} \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 = 1.15915, size = 358, normalized size = 6.75 \begin{align*} -\frac{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + b^{2} c^{2} d + a^{2} c d^{2} - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{3} + d^{4}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{3}} - \frac{3 \, b^{2} c^{2} e^{\left (4 \, x\right )} - 6 \, a b c d e^{\left (4 \, x\right )} + 3 \, a^{2} d^{2} e^{\left (4 \, x\right )} - 6 \, b^{2} c^{2} e^{\left (2 \, x\right )} + 12 \, a b c d e^{\left (2 \, x\right )} - 4 \, b^{2} c d e^{\left (2 \, x\right )} - 6 \, a^{2} d^{2} e^{\left (2 \, x\right )} + 8 \, a b d^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} d^{2} e^{\left (2 \, x\right )} + 3 \, b^{2} c^{2} - 6 \, a b c d + 4 \, b^{2} c d + 3 \, a^{2} d^{2} - 8 \, a b d^{2}}{2 \, d^{3}{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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