Optimal. Leaf size=64 \[ -\frac {\coth (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 64, 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, 2621, 321, 207} \[ -\frac {\coth (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2621
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\coth ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\cosh (e+f x) \int \text {csch}^2(e+f x) \text {sech}(e+f x) \, dx}{a \sqrt {a \cosh ^2(e+f x)}}\\ &=-\frac {(i \cosh (e+f x)) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,-i \text {csch}(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}}\\ &=-\frac {\coth (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}-\frac {(i \cosh (e+f x)) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}}\\ &=-\frac {\tan ^{-1}(\sinh (e+f x)) \cosh (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 46, normalized size = 0.72 \[ -\frac {\coth (e+f x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\sinh ^2(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 254, normalized size = 3.97 \[ -\frac {2 \, {\left ({\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (\cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )}\right )} \arctan \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + \cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \, {\left (a^{2} f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 51, normalized size = 0.80 \[ -\frac {\cosh \left (f x +e \right ) \left (\arctan \left (\sinh \left (f x +e \right )\right ) \sinh \left (f x +e \right )+1\right )}{a \sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 321, normalized size = 5.02 \[ -\frac {\frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} + 2 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}}}{2 \, f} - \frac {5 \, \sqrt {a} e^{\left (-f x - e\right )} + 6 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \, {\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} - 6 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 5 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \, {\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac {\arctan \left (e^{\left (-f x - e\right )}\right )}{2 \, a^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^2}{{\left (a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\coth ^{2}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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