Optimal. Leaf size=139 \[ -\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.181358, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3217, 1287, 1130, 208} \[ -\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3217
Rule 1287
Rule 1130
Rule 208
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2 \left (a-2 a x^2+(a-b) x^4\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}+\frac{b x^2}{a \left (a-2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x)}{a d}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=-\frac{\coth (c+d x)}{a d}+\frac{\left (\left (\sqrt{a}+\sqrt{b}\right ) \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a d}+\frac{\left (\left (1-\frac{\sqrt{a}}{\sqrt{b}}\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{-a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt{\sqrt{a}-\sqrt{b}} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt{\sqrt{a}+\sqrt{b}} d}-\frac{\coth (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.813646, size = 143, normalized size = 1.03 \[ \frac{\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}-a}}-2 \coth (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.059, size = 135, normalized size = 1. \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}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}-4\,a{{\it \_Z}}^{6}+ \left ( 6\,a-16\,b \right ){{\it \_Z}}^{4}-4\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}-{{\it \_R}}^{2}}{{{\it \_R}}^{7}a-3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{3}a-8\,{{\it \_R}}^{3}b-{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - 4 \, \int \frac{b e^{\left (6 \, d x + 6 \, c\right )} - 2 \, b e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )}}{a b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a b - 2 \,{\left (8 \, a^{2} e^{\left (4 \, c\right )} - 3 \, a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28628, size = 2840, normalized size = 20.43 \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 = 2.00349, size = 28, normalized size = 0.2 \begin{align*} -\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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