Optimal. Leaf size=66 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a f} \]
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Rubi [A] time = 0.129395, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3176, 3205, 16, 47, 63, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth ^3(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx &=\int \frac{\coth ^3(e+f x)}{\sqrt{a \cosh ^2(e+f x)}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(1-x)^2 \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a x}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a f}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a f}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cosh ^2(e+f x)}\right )}{2 a f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a f}\\ \end{align*}
Mathematica [A] time = 0.122417, size = 65, normalized size = 0.98 \[ -\frac{\cosh (e+f x) \left (\text{csch}^2\left (\frac{1}{2} (e+f x)\right )+\text{sech}^2\left (\frac{1}{2} (e+f x)\right )-4 \log \left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{8 f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.086, size = 42, normalized size = 0.6 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({( \left ( \sinh \left ( fx+e \right ) \right ) ^{-1}+ \left ( \sinh \left ( fx+e \right ) \right ) ^{-3}){\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.90444, size = 135, normalized size = 2.05 \begin{align*} -\frac{\log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, \sqrt{a} f} + \frac{\log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, \sqrt{a} f} + \frac{e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}}{{\left (2 \, \sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} - \sqrt{a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt{a}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94943, size = 1409, normalized size = 21.35 \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{\coth ^{3}{\left (e + f x \right )}}{\sqrt{a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36714, size = 109, normalized size = 1.65 \begin{align*} -\frac{\frac{\log \left (e^{\left (f x + e\right )} + 1\right )}{\sqrt{a}} - \frac{\log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )}{\sqrt{a}} + \frac{2 \,{\left (\sqrt{a} e^{\left (3 \, f x + 3 \, e\right )} + \sqrt{a} e^{\left (f x + e\right )}\right )}}{a{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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