Optimal. Leaf size=91 \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}-\frac{2 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.121694, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3176, 3207, 2590, 270} \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}-\frac{2 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \coth ^4(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt{a \cosh ^2(e+f x)} \coth ^4(e+f x) \, dx\\ &=\left (\sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \int \cosh (e+f x) \coth ^4(e+f x) \, dx\\ &=\frac{\left (i \sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^4} \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=\frac{\left (i \sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}-\frac{2}{x^2}\right ) \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac{2 \sqrt{a \cosh ^2(e+f x)} \text{csch}(e+f x) \text{sech}(e+f x)}{f}-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x)}{3 f}+\frac{\sqrt{a \cosh ^2(e+f x)} \tanh (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0729251, size = 47, normalized size = 0.52 \[ -\frac{\tanh (e+f x) \left (\text{csch}^4(e+f x)+6 \text{csch}^2(e+f x)-3\right ) \sqrt{a \cosh ^2(e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 55, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( fx+e \right ) a \left ( 3\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-6\, \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-1 \right ) }{3\, \left ( \sinh \left ( fx+e \right ) \right ) ^{3}f}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78277, size = 657, normalized size = 7.22 \begin{align*} -\frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right ) - \frac{2 \,{\left (9 \, \sqrt{a} e^{\left (-f x - e\right )} - 8 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}\right )}}{3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1}}{12 \, f} + \frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right ) + \frac{2 \,{\left (3 \, \sqrt{a} e^{\left (-f x - e\right )} - 8 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 9 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}\right )}}{3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1}}{12 \, f} + \frac{\sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )}}{f{\left (3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1\right )}} - \frac{33 \, \sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} - 40 \, \sqrt{a} e^{\left (-4 \, f x - 4 \, e\right )} + 15 \, \sqrt{a} e^{\left (-6 \, f x - 6 \, e\right )} - 6 \, \sqrt{a}}{12 \, f{\left (e^{\left (-f x - e\right )} - 3 \, e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, e^{\left (-5 \, f x - 5 \, e\right )} - e^{\left (-7 \, f x - 7 \, e\right )}\right )}} + \frac{15 \, \sqrt{a} e^{\left (-f x - e\right )} - 40 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 33 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )} - 6 \, \sqrt{a} e^{\left (-7 \, f x - 7 \, e\right )}}{12 \, f{\left (3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97043, size = 2361, normalized size = 25.95 \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 = 1.27173, size = 108, normalized size = 1.19 \begin{align*} -\frac{\sqrt{a}{\left (\frac{8 \,{\left (3 \, e^{\left (5 \, f x + 5 \, e\right )} - 4 \, e^{\left (3 \, f x + 3 \, e\right )} + 3 \, e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{3}} - 3 \, e^{\left (f x + e\right )} + 3 \, e^{\left (-f x - e\right )}\right )}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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