Optimal. Leaf size=69 \[ -\frac{2 \left (a c^2+b d^2\right ) \tanh ^{-1}\left (\frac{d-c \tanh \left (\frac{x}{2}\right )}{\sqrt{c^2+d^2}}\right )}{c^2 \sqrt{c^2+d^2}}+\frac{b d \tanh ^{-1}(\cosh (x))}{c^2}-\frac{b \coth (x)}{c} \]
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Rubi [A] time = 0.26344, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {4233, 3056, 3001, 3770, 2660, 618, 206} \[ -\frac{2 \left (a c^2+b d^2\right ) \tanh ^{-1}\left (\frac{d-c \tanh \left (\frac{x}{2}\right )}{\sqrt{c^2+d^2}}\right )}{c^2 \sqrt{c^2+d^2}}+\frac{b d \tanh ^{-1}(\cosh (x))}{c^2}-\frac{b \coth (x)}{c} \]
Antiderivative was successfully verified.
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Rule 4233
Rule 3056
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^2(x)}{c+d \sinh (x)} \, dx &=-\int \frac{\text{csch}^2(x) \left (-b-a \sinh ^2(x)\right )}{c+d \sinh (x)} \, dx\\ &=-\frac{b \coth (x)}{c}-\frac{i \int \frac{\text{csch}(x) (-i b d+i a c \sinh (x))}{c+d \sinh (x)} \, dx}{c}\\ &=-\frac{b \coth (x)}{c}-\frac{(b d) \int \text{csch}(x) \, dx}{c^2}+\left (a+\frac{b d^2}{c^2}\right ) \int \frac{1}{c+d \sinh (x)} \, dx\\ &=\frac{b d \tanh ^{-1}(\cosh (x))}{c^2}-\frac{b \coth (x)}{c}+\left (2 \left (a+\frac{b d^2}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+2 d x-c x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{b d \tanh ^{-1}(\cosh (x))}{c^2}-\frac{b \coth (x)}{c}-\left (4 \left (a+\frac{b d^2}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (c^2+d^2\right )-x^2} \, dx,x,2 d-2 c \tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{b d \tanh ^{-1}(\cosh (x))}{c^2}-\frac{2 \left (a+\frac{b d^2}{c^2}\right ) \tanh ^{-1}\left (\frac{d-c \tanh \left (\frac{x}{2}\right )}{\sqrt{c^2+d^2}}\right )}{\sqrt{c^2+d^2}}-\frac{b \coth (x)}{c}\\ \end{align*}
Mathematica [A] time = 0.410293, size = 125, normalized size = 1.81 \[ -\frac{\text{csch}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \left (\sinh (x) \left (b d \sqrt{-c^2-d^2} \log \left (\tanh \left (\frac{x}{2}\right )\right )-2 \left (a c^2+b d^2\right ) \tan ^{-1}\left (\frac{d-c \tanh \left (\frac{x}{2}\right )}{\sqrt{-c^2-d^2}}\right )\right )+b c \sqrt{-c^2-d^2} \cosh (x)\right )}{2 c^2 \sqrt{-c^2-d^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 112, normalized size = 1.6 \begin{align*} -{\frac{b}{2\,c}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{b}{2\,c} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{bd}{{c}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+2\,{\frac{a}{\sqrt{{c}^{2}+{d}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,c\tanh \left ( x/2 \right ) -2\,d}{\sqrt{{c}^{2}+{d}^{2}}}} \right ) }+2\,{\frac{b{d}^{2}}{{c}^{2}\sqrt{{c}^{2}+{d}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,c\tanh \left ( x/2 \right ) -2\,d}{\sqrt{{c}^{2}+{d}^{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 = 6.54043, size = 1037, normalized size = 15.03 \begin{align*} \frac{2 \, b c^{3} + 2 \, b c d^{2} +{\left (a c^{2} + b d^{2} -{\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a c^{2} + b d^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt{c^{2} + d^{2}} \log \left (\frac{d^{2} \cosh \left (x\right )^{2} + d^{2} \sinh \left (x\right )^{2} + 2 \, c d \cosh \left (x\right ) + 2 \, c^{2} + d^{2} + 2 \,{\left (d^{2} \cosh \left (x\right ) + c d\right )} \sinh \left (x\right ) - 2 \, \sqrt{c^{2} + d^{2}}{\left (d \cosh \left (x\right ) + d \sinh \left (x\right ) + c\right )}}{d \cosh \left (x\right )^{2} + d \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \,{\left (d \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) - d}\right ) +{\left (b c^{2} d + b d^{3} -{\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (b c^{2} d + b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (b c^{2} d + b d^{3} -{\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (b c^{2} d + b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{c^{4} + c^{2} d^{2} -{\left (c^{4} + c^{2} d^{2}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (c^{4} + c^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (c^{4} + c^{2} d^{2}\right )} \sinh \left (x\right )^{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{a + b \operatorname{csch}^{2}{\left (x \right )}}{c + d \sinh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17998, size = 147, normalized size = 2.13 \begin{align*} \frac{b d \log \left (e^{x} + 1\right )}{c^{2}} - \frac{b d \log \left ({\left | e^{x} - 1 \right |}\right )}{c^{2}} + \frac{{\left (a c^{2} + b d^{2}\right )} \log \left (\frac{{\left | 2 \, d e^{x} + 2 \, c - 2 \, \sqrt{c^{2} + d^{2}} \right |}}{{\left | 2 \, d e^{x} + 2 \, c + 2 \, \sqrt{c^{2} + d^{2}} \right |}}\right )}{\sqrt{c^{2} + d^{2}} c^{2}} - \frac{2 \, b}{c{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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