Optimal. Leaf size=37 \[ -\frac {2 \text {csch}(a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2670, 276}
\begin {gather*} \frac {\sinh (a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}-\frac {2 \text {csch}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 276
Rule 2670
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth ^4(a+b x) \, dx &=\frac {i \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac {i \text {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac {2 \text {csch}(a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} -\frac {2 \text {csch}(a+b x)}{b}-\frac {\text {csch}^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.03, size = 51, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\frac {\cosh ^{4}\left (b x +a \right )}{\sinh \left (b x +a \right )^{3}}-\frac {4 \left (\cosh ^{2}\left (b x +a \right )\right )}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(51\) |
default | \(\frac {\frac {\cosh ^{4}\left (b x +a \right )}{\sinh \left (b x +a \right )^{3}}-\frac {4 \left (\cosh ^{2}\left (b x +a \right )\right )}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(51\) |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}-\frac {4 \,{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a}-4 \,{\mathrm e}^{2 b x +2 a}+3\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (35) = 70\).
time = 0.26, size = 100, normalized size = 2.70 \begin {gather*} -\frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {33 \, e^{\left (-2 \, b x - 2 \, a\right )} - 41 \, e^{\left (-4 \, b x - 4 \, a\right )} + 27 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3}{6 \, b {\left (e^{\left (-b x - a\right )} - 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} - e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs.
\(2 (35) = 70\).
time = 0.40, size = 89, normalized size = 2.41 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right )^{4} + 3 \, \sinh \left (b x + a\right )^{4} + 18 \, {\left (\cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{2} - 36 \, \cosh \left (b x + a\right )^{2} + 25}{6 \, {\left (b \sinh \left (b x + a\right )^{3} + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\left (a + b x \right )} \coth ^{4}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (35) = 70\).
time = 0.40, size = 71, normalized size = 1.92 \begin {gather*} -\frac {\frac {8 \, {\left (3 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 2\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} - 3 \, e^{\left (b x + a\right )} + 3 \, e^{\left (-b x - a\right )}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.45, size = 131, normalized size = 3.54 \begin {gather*} \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________