Optimal. Leaf size=29 \[ \frac {\sinh (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3190, 191} \[ \frac {\sinh (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cosh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\sinh (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.00 \[ \frac {\sinh (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 245, normalized size = 8.45 \[ \frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} - 1\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{a b f \cosh \left (f x + e\right )^{4} + 4 \, a b f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a b f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{2} - a b\right )} f \cosh \left (f x + e\right )^{2} + a b f + 2 \, {\left (3 \, a b f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{2} - a b\right )} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a b f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} - a b\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \]
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 [A] time = 0.03, size = 28, normalized size = 0.97 \[ \frac {\sinh \left (f x +e \right )}{a f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 236, normalized size = 8.14 \[ \frac {b^{2} e^{\left (-6 \, f x - 6 \, e\right )} + 2 \, a b - b^{2} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} - \frac {b^{2} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + {\left (2 \, a b - b^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 191, normalized size = 6.59 \[ -\frac {{\mathrm {e}}^{e+f\,x}\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}\,\left (\frac {2\,\mathrm {cosh}\left (e+f\,x\right )\,{\mathrm {e}}^{e+f\,x}\,\left (b\,\left (2\,a-b\right )-b\,\left (4\,a-2\,b\right )\right )}{f\,\left (a\,b^2-a^2\,b\right )}-\frac {2\,b^2\,{\mathrm {e}}^{e+f\,x}\,\mathrm {sinh}\left (e+f\,x\right )}{f\,\left (a\,b^2-a^2\,b\right )}+\frac {b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\left (4\,a-2\,b\right )}{f\,\left (a\,b^2-a^2\,b\right )}\right )}{4\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}-2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )} \]
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 \[ \int \frac {\cosh {\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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