Optimal. Leaf size=66 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a^2 f} \]
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Rubi [A]
time = 0.10, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3255, 3284, 16,
44, 65, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\text {csch}^2(e+f x) \sqrt {a \cosh ^2(e+f x)}}{2 a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 44
Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\coth ^3(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\coth ^3(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x}{(1-x)^2 (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{(1-x)^2 \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a^2 f}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 a f}\\ &=-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a^2 f}+\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(e+f x)}\right )}{2 a^2 f}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 67, normalized size = 1.02 \begin {gather*} -\frac {\cosh ^3(e+f x) \left (\text {csch}^2\left (\frac {1}{2} (e+f x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (e+f x)\right )\right )}{8 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \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 = 1.20, size = 36, normalized size = 0.55
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {1}{\sinh \left (f x +e \right )^{3} a \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(36\) |
risch | \(-\frac {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2}}{a \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{f x}+{\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{2 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}-\frac {\ln \left ({\mathrm e}^{f x}-{\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{2 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 106, normalized size = 1.61 \begin {gather*} \frac {e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}}{{\left (2 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} - a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - a^{\frac {3}{2}}\right )} f} + \frac {\log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, a^{\frac {3}{2}} f} - \frac {\log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, a^{\frac {3}{2}} 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. 565 vs.
\(2 (54) = 108\).
time = 0.60, size = 565, normalized size = 8.56 \begin {gather*} -\frac {{\left (6 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 2 \, e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 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 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} e^{\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 )} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}\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 (a^{2} f \cosh \left (f x + e\right )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\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 )^{3} + a^{2} f + 2 \, {\left (3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f + {\left (3 \, 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 )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, {\left (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 )^{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 )}} \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 {\coth ^{3}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^3}{{\left (a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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