Optimal. Leaf size=21 \[ -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3176, 3205, 16, 32} \[ -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 32
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \frac {\tanh (e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tanh (e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{(a x)^{5/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 21, normalized size = 1.00 \[ -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 608, normalized size = 28.95 \[ -\frac {8 \, {\left (\cosh \left (f x + e\right )^{3} e^{\left (f x + e\right )} + 3 \, \cosh \left (f x + e\right )^{2} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + 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}\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 )}}{3 \, {\left (a^{2} f \cosh \left (f x + e\right )^{6} + 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{6} + 6 \, {\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 )^{5} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} + 3 \, a^{2} f \cosh \left (f x + e\right ) + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} + 3 \, a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{3} + a^{2} f + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} + 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} + 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{6} + 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right )^{5} + 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right ) + {\left (a^{2} f \cosh \left (f x + e\right )^{5} + 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 32, normalized size = 1.52 \[ -\frac {8 \, e^{\left (3 \, f x + 3 \, e\right )}}{3 \, a^{\frac {3}{2}} f {\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 20, normalized size = 0.95 \[ -\frac {1}{3 f \left (a +a \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 61, normalized size = 2.90 \[ -\frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (3 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 58, normalized size = 2.76 \[ -\frac {16\,{\mathrm {e}}^{4\,e+4\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4} \]
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 \[ \int \frac {\tanh {\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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