Optimal. Leaf size=62 \[ \frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{2 f \sqrt {a \cosh ^2(e+f x)}}-\frac {\tanh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3176, 3207, 2611, 3770} \[ \frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{2 f \sqrt {a \cosh ^2(e+f x)}}-\frac {\tanh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh ^2(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx &=\int \frac {\tanh ^2(e+f x)}{\sqrt {a \cosh ^2(e+f x)}} \, dx\\ &=\frac {\cosh (e+f x) \int \text {sech}(e+f x) \tanh ^2(e+f x) \, dx}{\sqrt {a \cosh ^2(e+f x)}}\\ &=-\frac {\tanh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}}+\frac {\cosh (e+f x) \int \text {sech}(e+f x) \, dx}{2 \sqrt {a \cosh ^2(e+f x)}}\\ &=\frac {\tan ^{-1}(\sinh (e+f x)) \cosh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}}-\frac {\tanh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 44, normalized size = 0.71 \[ \frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))-\tanh (e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 504, normalized size = 8.13 \[ -\frac {{\left (3 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + {\left (3 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) - {\left (4 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (\cosh \left (f x + e\right )^{4} + 2 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )}\right )} \arctan \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + {\left (\cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )}\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{a f \cosh \left (f x + e\right )^{4} + {\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{4} + 2 \, a f \cosh \left (f x + e\right )^{2} + 4 \, {\left (a f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (3 \, a f \cosh \left (f x + e\right )^{2} + a f + {\left (3 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + a f + {\left (a f \cosh \left (f x + e\right )^{4} + 2 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, {\left (a f \cosh \left (f x + e\right )^{3} + a f \cosh \left (f x + e\right ) + {\left (a f \cosh \left (f x + e\right )^{3} + a f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 63, normalized size = 1.02 \[ \frac {\frac {\arctan \left (e^{\left (f x + e\right )}\right )}{\sqrt {a}} - \frac {\sqrt {a} e^{\left (3 \, f x + 3 \, e\right )} - \sqrt {a} e^{\left (f x + e\right )}}{a {\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 51, normalized size = 0.82 \[ \frac {\frac {\arctan \left (\sinh \left (f x +e \right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}-\frac {\sinh \left (f x +e \right )}{2}}{\cosh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 217, normalized size = 3.50 \[ \frac {\frac {\arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} - \frac {e^{\left (-f x - e\right )} - e^{\left (-3 \, f x - 3 \, e\right )}}{2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a}}}{2 \, f} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{2 \, \sqrt {a} f} - \frac {5 \, e^{\left (-f x - e\right )} + 3 \, e^{\left (-3 \, f x - 3 \, e\right )}}{4 \, {\left (2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a}\right )} f} + \frac {3 \, e^{\left (-f x - e\right )} + 5 \, e^{\left (-3 \, f x - 3 \, e\right )}}{4 \, {\left (2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^2}{\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\tanh ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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