Optimal. Leaf size=57 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} d \sqrt{a-b}}-\frac{\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.0808976, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3187, 453, 208} \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} d \sqrt{a-b}}-\frac{\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 453
Rule 208
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{x^2 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x)}{a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} \sqrt{a-b} d}-\frac{\coth (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.292172, size = 57, normalized size = 1. \[ \frac{-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a-b}}-\sqrt{a} \coth (c+d x)}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 319, normalized size = 5.6 \begin{align*} -{\frac{1}{2\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{b}{da}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}+{\frac{{b}^{2}}{da}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}+{\frac{b}{da}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}}+{\frac{{b}^{2}}{da}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \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 = 2.21384, size = 1661, normalized size = 29.14 \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 = 1.30799, size = 100, normalized size = 1.75 \begin{align*} -\frac{b \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} a d} - \frac{2}{a d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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