Optimal. Leaf size=38 \[ \frac {a}{f \sqrt {a \cosh ^2(e+f x)}}+\frac {\sqrt {a \cosh ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.11, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3176, 3205, 16, 43} \[ \frac {a}{f \sqrt {a \cosh ^2(e+f x)}}+\frac {\sqrt {a \cosh ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 43
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^3(e+f x) \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(1-x) \sqrt {a x}}{x^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {1-x}{(a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {1}{(a x)^{3/2}}-\frac {1}{a \sqrt {a x}}\right ) \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {a}{f \sqrt {a \cosh ^2(e+f x)}}+\frac {\sqrt {a \cosh ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 29, normalized size = 0.76 \[ \frac {a \left (\cosh ^2(e+f x)+1\right )}{f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 311, normalized size = 8.18 \[ \frac {{\left (4 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 6 \, {\left (\cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} + 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (\cosh \left (f x + e\right )^{4} + 6 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )}\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \, {\left (f \cosh \left (f x + e\right )^{3} + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{2} + f \cosh \left (f x + e\right ) + {\left (f \cosh \left (f x + e\right )^{3} + f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (3 \, f \cosh \left (f x + e\right )^{2} + {\left (3 \, f \cosh \left (f x + e\right )^{2} + f\right )} e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 53, normalized size = 1.39 \[ \frac {\sqrt {a} {\left (\frac {{\left (5 \, e^{\left (2 \, f x + 2 \, e\right )} + 1\right )} e^{\left (-e\right )}}{e^{\left (3 \, f x + 2 \, e\right )} + e^{\left (f x\right )}} + e^{\left (f x + e\right )}\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 42, normalized size = 1.11 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\left (\sinh ^{3}\left (f x +e \right )\right ) a}{\cosh \left (f x +e \right )^{2} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 106, normalized size = 2.79 \[ \frac {3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}\right )}} + \frac {\sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )}}{2 \, f {\left (e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}\right )}} + \frac {\sqrt {a}}{2 \, f {\left (e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 67, normalized size = 1.76 \[ \frac {\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}\,\left (6\,{\mathrm {e}}^{2\,e+2\,f\,x}+{\mathrm {e}}^{4\,e+4\,f\,x}+1\right )}{f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^2} \]
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 \[ \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \tanh ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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