Optimal. Leaf size=61 \[ \frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {3 \sinh (x) \tanh ^{-1}(\cosh (x))}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3204, 3207, 3770} \[ \frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {3 \sinh (x) \tanh ^{-1}(\cosh (x))}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sinh ^2(x)\right )^{5/2}} \, dx &=-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}-\frac {3 \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx}{8 a^2}\\ &=-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}+\frac {(3 \sinh (x)) \int \text {csch}(x) \, dx}{8 a^2 \sqrt {a \sinh ^2(x)}}\\ &=-\frac {\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac {3 \coth (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}-\frac {3 \tanh ^{-1}(\cosh (x)) \sinh (x)}{8 a^2 \sqrt {a \sinh ^2(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 67, normalized size = 1.10 \[ -\frac {\text {csch}(x) \sqrt {a \sinh ^2(x)} \left (\text {csch}^4\left (\frac {x}{2}\right )-6 \text {csch}^2\left (\frac {x}{2}\right )-\text {sech}^4\left (\frac {x}{2}\right )-6 \text {sech}^2\left (\frac {x}{2}\right )-24 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{64 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.57, size = 875, normalized size = 14.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 52, normalized size = 0.85 \[ \frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2} a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 89, normalized size = 1.46 \[ \frac {\sqrt {a \left (\cosh ^{2}\relax (x )\right )}\, \left (-3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}+2 a}{\sinh \relax (x )}\right ) a \left (\sinh ^{4}\relax (x )\right )+3 \sqrt {a \left (\cosh ^{2}\relax (x )\right )}\, \left (\sinh ^{2}\relax (x )\right ) \sqrt {a}-2 \sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}\right )}{8 a^{\frac {7}{2}} \sinh \relax (x )^{3} \cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 96, normalized size = 1.57 \[ \frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 6 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 4 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} - a^{\frac {5}{2}} e^{\left (-8 \, x\right )} - a^{\frac {5}{2}}\right )}} + \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a^{\frac {5}{2}}} - \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a\,{\mathrm {sinh}\relax (x)}^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sinh ^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________