Optimal. Leaf size=29 \[ \frac {2}{\sqrt {1-\sinh ^2(x)}}+2 \sqrt {1-\sinh ^2(x)} \]
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Rubi [A]
time = 0.07, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 272, 45}
\begin {gather*} 2 \sqrt {1-\sinh ^2(x)}+\frac {2}{\sqrt {1-\sinh ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx &=i \text {Subst}\left (\int -\frac {2 i x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right )\\ &=2 \text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right )\\ &=\text {Subst}\left (\int \frac {x}{(1-x)^{3/2}} \, dx,x,\sinh ^2(x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{(1-x)^{3/2}}-\frac {1}{\sqrt {1-x}}\right ) \, dx,x,\sinh ^2(x)\right )\\ &=\frac {2}{\sqrt {1-\sinh ^2(x)}}+2 \sqrt {1-\sinh ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 21, normalized size = 0.72 \begin {gather*} \frac {5-\cosh (2 x)}{\sqrt {1-\sinh ^2(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 = 0.11, size = 28, normalized size = 0.97
method | result | size |
default | \(\mathit {`\,int/indef0`\,}\left (-\frac {2 \left (\sinh ^{3}\left (x \right )\right )}{\left (\sinh ^{2}\left (x \right )-1\right ) \sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}, \sinh \left (x \right )\right )\) | \(28\) |
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. 177 vs.
\(2 (25) = 50\).
time = 1.70, size = 177, normalized size = 6.10 \begin {gather*} -\frac {16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \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. 161 vs.
\(2 (25) = 50\).
time = 0.72, size = 161, normalized size = 5.55 \begin {gather*} \frac {\sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 2 \, {\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{3} + 2 \, {\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\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 {\sinh ^{2}{\left (x \right )} \sinh {\left (2 x \right )}}{\left (- \left (\sinh {\left (x \right )} - 1\right ) \left (\sinh {\left (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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 47, normalized size = 1.62 \begin {gather*} \frac {2\,\sqrt {1-{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,x}-10\,{\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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