Optimal. Leaf size=66 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {\text {csch}^2(e+f x) \sqrt {a \cosh ^2(e+f x)}}{2 a f} \]
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Rubi [A] time = 0.13, 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 = {3176, 3205, 16, 47, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {\text {csch}^2(e+f x) \sqrt {a \cosh ^2(e+f x)}}{2 a f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 47
Rule 63
Rule 206
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx &=\int \frac {\coth ^3(e+f x)}{\sqrt {a \cosh ^2(e+f x)}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(1-x)^2 \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a x}}{(1-x)^2} \, 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 f}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a f}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(e+f x)}\right )}{2 a f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a f}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 65, normalized size = 0.98 \[ -\frac {\cosh (e+f x) \left (\text {csch}^2\left (\frac {1}{2} (e+f x)\right )+\text {sech}^2\left (\frac {1}{2} (e+f x)\right )-4 \log \left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{8 f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 529, normalized size = 8.02 \[ -\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 f \cosh \left (f x + e\right )^{4} + {\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + 4 \, {\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 )^{3} + 2 \, {\left (3 \, a f \cosh \left (f x + e\right )^{2} - a f + {\left (3 \, 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 )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, {\left (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 )^{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 )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 42, normalized size = 0.64 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\frac {1}{\sinh \left (f x +e \right )}+\frac {1}{\sinh \left (f x +e \right )^{3}}}{\sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.62, size = 100, normalized size = 1.52 \[ -\frac {\log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, \sqrt {a} f} + \frac {\log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, \sqrt {a} f} + \frac {e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}}{{\left (2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} - \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt {a}\right )} f} \]
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 \[ \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^3}{\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\coth ^{3}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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