Optimal. Leaf size=17 \[ -\frac {1}{6} \text {csch}^6(x)-\frac {\text {csch}^4(x)}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2606, 14} \[ -\frac {1}{6} \text {csch}^6(x)-\frac {\text {csch}^4(x)}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2606
Rubi steps
\begin {align*} \int \coth ^3(x) \text {csch}^4(x) \, dx &=\operatorname {Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,-i \text {csch}(x)\right )\\ &=\operatorname {Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,-i \text {csch}(x)\right )\\ &=-\frac {1}{4} \text {csch}^4(x)-\frac {\text {csch}^6(x)}{6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {1}{6} \text {csch}^6(x)-\frac {\text {csch}^4(x)}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 222, normalized size = 13.06 \[ -\frac {4 \, {\left (3 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x) \sinh \relax (x)^{3} + 3 \, \sinh \relax (x)^{4} + 2 \, {\left (9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 3\right )}}{3 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 2 \, {\left (14 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{6} - 6 \, \cosh \relax (x)^{6} + 4 \, {\left (14 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} - 45 \, \cosh \relax (x)^{2} + 8\right )} \sinh \relax (x)^{4} + 16 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} - 15 \, \cosh \relax (x)^{3} + 7 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (14 \, \cosh \relax (x)^{6} - 45 \, \cosh \relax (x)^{4} + 48 \, \cosh \relax (x)^{2} - 13\right )} \sinh \relax (x)^{2} - 26 \, \cosh \relax (x)^{2} + 4 \, {\left (2 \, \cosh \relax (x)^{7} - 9 \, \cosh \relax (x)^{5} + 14 \, \cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 15\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 29, normalized size = 1.71 \[ -\frac {4 \, {\left (3 \, e^{\left (8 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )}\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 18, normalized size = 1.06 \[ -\frac {\cosh ^{2}\relax (x )}{4 \sinh \relax (x )^{6}}+\frac {1}{12 \sinh \relax (x )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 139, normalized size = 8.18 \[ \frac {4 \, e^{\left (-4 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} + \frac {8 \, e^{\left (-6 \, x\right )}}{3 \, {\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} + \frac {4 \, e^{\left (-8 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.48, size = 210, normalized size = 12.35 \[ -\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {12\,{\mathrm {e}}^{4\,x}}{5}+\frac {16\,{\mathrm {e}}^{6\,x}}{15}+\frac {4}{15}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {\frac {4\,{\mathrm {e}}^{2\,x}}{3}+4\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+\frac {4\,{\mathrm {e}}^{8\,x}}{3}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{15}+\frac {2}{5}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {4}{15\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {6\,{\mathrm {e}}^{2\,x}}{5}+\frac {4\,{\mathrm {e}}^{4\,x}}{5}+\frac {2}{5}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{3}{\relax (x )} \operatorname {csch}^{4}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________