Optimal. Leaf size=106 \[ \frac {\tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\tanh (e+f x) \text {sech}^2(e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3176, 3207, 2611, 3768, 3770} \[ \frac {\tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\tanh (e+f x) \text {sech}^2(e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3176
Rule 3207
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tanh ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\cosh (e+f x) \int \text {sech}^3(e+f x) \tanh ^2(e+f x) \, dx}{a \sqrt {a \cosh ^2(e+f x)}}\\ &=-\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\cosh (e+f x) \int \text {sech}^3(e+f x) \, dx}{4 a \sqrt {a \cosh ^2(e+f x)}}\\ &=\frac {\tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\cosh (e+f x) \int \text {sech}(e+f x) \, dx}{8 a \sqrt {a \cosh ^2(e+f x)}}\\ &=\frac {\tan ^{-1}(\sinh (e+f x)) \cosh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 58, normalized size = 0.55 \[ \frac {\tanh (e+f x) \left (1-2 \text {sech}^2(e+f x)\right )+\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 a f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1423, normalized size = 13.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 93, normalized size = 0.88 \[ \frac {\frac {\arctan \left (e^{\left (f x + e\right )}\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {a} e^{\left (7 \, f x + 7 \, e\right )} - 7 \, \sqrt {a} e^{\left (5 \, f x + 5 \, e\right )} + 7 \, \sqrt {a} e^{\left (3 \, f x + 3 \, e\right )} - \sqrt {a} e^{\left (f x + e\right )}}{a^{2} {\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{4}}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 69, normalized size = 0.65 \[ \frac {\arctan \left (\sinh \left (f x +e \right )\right ) \left (\cosh ^{4}\left (f x +e \right )\right )+\left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )-2 \sinh \left (f x +e \right )}{8 a \cosh \left (f x +e \right )^{3} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 369, normalized size = 3.48 \[ -\frac {\frac {3 \, e^{\left (-f x - e\right )} + 11 \, e^{\left (-3 \, f x - 3 \, e\right )} - 11 \, e^{\left (-5 \, f x - 5 \, e\right )} - 3 \, e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}}}{8 \, f} + \frac {15 \, e^{\left (-f x - e\right )} + 55 \, e^{\left (-3 \, f x - 3 \, e\right )} + 73 \, e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, e^{\left (-7 \, f x - 7 \, e\right )}}{48 \, {\left (4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {15 \, e^{\left (-f x - e\right )} - 73 \, e^{\left (-3 \, f x - 3 \, e\right )} - 55 \, e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, e^{\left (-7 \, f x - 7 \, e\right )}}{48 \, {\left (4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {5 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{8 \, a^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^2}{{\left (a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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