Optimal. Leaf size=17 \[ -\frac{1}{6} \text{csch}^6(x)-\frac{\text{csch}^4(x)}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0280708, antiderivative size = 17, normalized size of antiderivative = 1., 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 2606
Rule 14
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 = 0.0090641, size = 17, normalized size = 1. \[ -\frac{1}{6} \text{csch}^6(x)-\frac{\text{csch}^4(x)}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 32, normalized size = 1.9 \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{6\, \left ( \sinh \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{12\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{12\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02277, size = 188, normalized size = 11.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.70895, size = 737, normalized size = 43.35 \begin{align*} -\frac{4 \,{\left (3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \,{\left (9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{3 \,{\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \,{\left (14 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 6 \, \cosh \left (x\right )^{6} + 4 \,{\left (14 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} - 45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{4} + 16 \, \cosh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} - 15 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \,{\left (14 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} + 48 \, \cosh \left (x\right )^{2} - 13\right )} \sinh \left (x\right )^{2} - 26 \, \cosh \left (x\right )^{2} + 4 \,{\left (2 \, \cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} + 14 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 15\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18309, size = 39, normalized size = 2.29 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]