Optimal. Leaf size=31 \[ -\frac {\cosh ^3(a+b x)}{3 b}+\frac {\cosh ^5(a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2645, 14}
\begin {gather*} \frac {\cosh ^5(a+b x)}{5 b}-\frac {\cosh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2645
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=-\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\cosh ^3(a+b x)}{3 b}+\frac {\cosh ^5(a+b x)}{5 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 27, normalized size = 0.87 \begin {gather*} \frac {\cosh ^3(a+b x) (-7+3 \cosh (2 (a+b x)))}{30 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 34, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}}{b}\) | \(34\) |
default | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}}{b}\) | \(34\) |
risch | \(\frac {{\mathrm e}^{5 b x +5 a}}{160 b}-\frac {{\mathrm e}^{3 b x +3 a}}{96 b}-\frac {{\mathrm e}^{b x +a}}{16 b}-\frac {{\mathrm e}^{-b x -a}}{16 b}-\frac {{\mathrm e}^{-3 b x -3 a}}{96 b}+\frac {{\mathrm e}^{-5 b x -5 a}}{160 b}\) | \(83\) |
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. 78 vs.
\(2 (27) = 54\).
time = 0.26, size = 78, normalized size = 2.52 \begin {gather*} -\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 30 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3\right )} e^{\left (5 \, b x + 5 \, a\right )}}{480 \, b} - \frac {30 \, e^{\left (-b x - a\right )} + 5 \, e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{480 \, b} \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. 79 vs.
\(2 (27) = 54\).
time = 0.41, size = 79, normalized size = 2.55 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right )^{5} + 15 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 5 \, \cosh \left (b x + a\right )^{3} + 15 \, {\left (2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 30 \, \cosh \left (b x + a\right )}{240 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.27, size = 44, normalized size = 1.42 \begin {gather*} \begin {cases} \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 \cosh ^{5}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \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. 82 vs.
\(2 (27) = 54\).
time = 0.39, size = 82, normalized size = 2.65 \begin {gather*} \frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} - \frac {e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} - \frac {e^{\left (b x + a\right )}}{16 \, b} - \frac {e^{\left (-b x - a\right )}}{16 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} + \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.45, size = 26, normalized size = 0.84 \begin {gather*} -\frac {5\,{\mathrm {cosh}\left (a+b\,x\right )}^3-3\,{\mathrm {cosh}\left (a+b\,x\right )}^5}{15\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________