Optimal. Leaf size=87 \[ \frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}-\frac {\text {csch}^2(e+f x) \left (a \cosh ^2(e+f x)\right )^{3/2}}{2 a f} \]
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Rubi [A] time = 0.14, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3176, 3205, 16, 47, 50, 63, 206} \[ \frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}-\frac {\text {csch}^2(e+f x) \left (a \cosh ^2(e+f x)\right )^{3/2}}{2 a f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 47
Rule 50
Rule 63
Rule 206
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \coth ^3(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \coth ^3(e+f x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \sqrt {a x}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a x)^{3/2}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {a x}}{1-x} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(e+f x)}\right )}{2 f}\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}+\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 77, normalized size = 0.89 \[ \frac {\text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \left (8 \cosh (e+f x)-\text {csch}^2\left (\frac {1}{2} (e+f x)\right )-\text {sech}^2\left (\frac {1}{2} (e+f x)\right )+12 \log \left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{8 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.39, size = 764, normalized size = 8.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 89, normalized size = 1.02 \[ -\frac {\sqrt {a} {\left (\frac {2 \, {\left (e^{\left (3 \, f x + 3 \, e\right )} + e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{2}} - e^{\left (f x + e\right )} - e^{\left (-f x - e\right )} + 3 \, \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, \log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 54, normalized size = 0.62 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {a \left (\cosh ^{4}\left (f x +e \right )\right )}{\sinh \left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )-1\right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 126, normalized size = 1.45 \[ -\frac {3 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, f} + \frac {3 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, f} - \frac {3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} - \sqrt {a}}{2 \, f {\left (e^{\left (-f x - e\right )} - 2 \, e^{\left (-3 \, f x - 3 \, e\right )} + e^{\left (-5 \, f x - 5 \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {coth}\left (e+f\,x\right )}^3\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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