Optimal. Leaf size=42 \[ \frac {a}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}-\frac {1}{f \sqrt {a \cosh ^2(e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 42, 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 = {3255, 3284, 16,
45} \begin {gather*} \frac {a}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}-\frac {1}{f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\tanh ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx &=\int \frac {\tanh ^3(e+f x)}{\sqrt {a \cosh ^2(e+f x)}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {1-x}{x^2 \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \frac {1-x}{(a x)^{5/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{(a x)^{5/2}}-\frac {1}{a (a x)^{3/2}}\right ) \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {a}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}-\frac {1}{f \sqrt {a \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 31, normalized size = 0.74 \begin {gather*} \frac {-3+\text {sech}^2(e+f x)}{3 f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.12, size = 41, normalized size = 0.98
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh ^{3}\left (f x +e \right )}{\cosh \left (f x +e \right )^{4} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(41\) |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{4 f x +4 e}+2 \,{\mathrm e}^{2 f x +2 e}+3\right )}{3 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} f}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs.
\(2 (38) = 76\).
time = 0.52, size = 196, normalized size = 4.67 \begin {gather*} -\frac {2 \, e^{\left (-f x - e\right )}}{{\left (3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + \sqrt {a}\right )} f} - \frac {4 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + \sqrt {a}\right )} f} - \frac {2 \, e^{\left (-5 \, f x - 5 \, e\right )}}{{\left (3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + \sqrt {a}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 641 vs.
\(2 (36) = 72\).
time = 0.47, size = 641, normalized size = 15.26 \begin {gather*} -\frac {2 \, {\left (15 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 3 \, e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + 2 \, {\left (15 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 6 \, {\left (5 \, \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 )^{2} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{4} + 2 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (3 \, \cosh \left (f x + e\right )^{5} + 2 \, \cosh \left (f x + e\right )^{3} + 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 )}}{3 \, {\left (a f \cosh \left (f x + e\right )^{6} + {\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{6} + 3 \, a f \cosh \left (f x + e\right )^{4} + 6 \, {\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 )^{5} + 3 \, {\left (5 \, a f \cosh \left (f x + e\right )^{2} + a f + {\left (5 \, 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 )^{4} + 3 \, a f \cosh \left (f x + e\right )^{2} + 4 \, {\left (5 \, a f \cosh \left (f x + e\right )^{3} + 3 \, a f \cosh \left (f x + e\right ) + {\left (5 \, a f \cosh \left (f x + e\right )^{3} + 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 )^{3} + 3 \, {\left (5 \, a f \cosh \left (f x + e\right )^{4} + 6 \, a f \cosh \left (f x + e\right )^{2} + a f + {\left (5 \, a f \cosh \left (f x + e\right )^{4} + 6 \, 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 )^{6} + 3 \, a f \cosh \left (f x + e\right )^{4} + 3 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a f \cosh \left (f x + e\right )^{5} + 2 \, 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 )^{5} + 2 \, 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 )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\tanh ^{3}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 82, normalized size = 1.95 \begin {gather*} -\frac {4\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}\,\left (2\,{\mathrm {e}}^{2\,e+2\,f\,x}+3\,{\mathrm {e}}^{4\,e+4\,f\,x}+3\right )}{3\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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