Optimal. Leaf size=40 \[ \frac {3}{8} \sin ^{-1}(\coth (x))+\frac {1}{4} \coth (x) \left (-\text {csch}^2(x)\right )^{3/2}+\frac {3}{8} \coth (x) \sqrt {-\text {csch}^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 40, 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 = {4122, 195, 216} \[ \frac {3}{8} \sin ^{-1}(\coth (x))+\frac {1}{4} \coth (x) \left (-\text {csch}^2(x)\right )^{3/2}+\frac {3}{8} \coth (x) \sqrt {-\text {csch}^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 216
Rule 4122
Rubi steps
\begin {align*} \int \left (-\text {csch}^2(x)\right )^{5/2} \, dx &=\operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{4} \coth (x) \left (-\text {csch}^2(x)\right )^{3/2}+\frac {3}{4} \operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {3}{8} \coth (x) \sqrt {-\text {csch}^2(x)}+\frac {1}{4} \coth (x) \left (-\text {csch}^2(x)\right )^{3/2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\coth (x)\right )\\ &=\frac {3}{8} \sin ^{-1}(\coth (x))+\frac {3}{8} \coth (x) \sqrt {-\text {csch}^2(x)}+\frac {1}{4} \coth (x) \left (-\text {csch}^2(x)\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 41, normalized size = 1.02 \[ \frac {1}{64} \sinh (x) \left (-\text {csch}^2(x)\right )^{5/2} \left (6 \left (\cosh (3 x)+4 \sinh ^4(x) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )-22 \cosh (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.50, size = 115, normalized size = 2.88 \[ \frac {{\left (-3 i \, e^{\left (8 \, x\right )} + 12 i \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (4 \, x\right )} + 12 i \, e^{\left (2 \, x\right )} - 3 i\right )} \log \left (e^{x} + 1\right ) + {\left (3 i \, e^{\left (8 \, x\right )} - 12 i \, e^{\left (6 \, x\right )} + 18 i \, e^{\left (4 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} + 3 i\right )} \log \left (e^{x} - 1\right ) + 6 i \, e^{\left (7 \, x\right )} - 22 i \, e^{\left (5 \, x\right )} - 22 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x}}{8 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.14, size = 72, normalized size = 1.80 \[ -\frac {1}{16} \, {\left (\frac {4 i \, {\left (3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} - 3 i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + 3 i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.22, size = 114, normalized size = 2.85 \[ \frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}+3\right )}{4 \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {3 \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right )}{8}+\frac {3 \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.52, size = 74, normalized size = 1.85 \[ \frac {3 i \, e^{\left (-x\right )} - 11 i \, e^{\left (-3 \, x\right )} - 11 i \, e^{\left (-5 \, x\right )} + 3 i \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {3}{8} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{8} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (-\frac {1}{{\mathrm {sinh}\relax (x)}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________