Optimal. Leaf size=144 \[ -\frac {6 b^2 d e^{c+d x} \cosh (a+b x)}{9 b^4-10 b^2 d^2+d^4}-\frac {d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac {6 b^3 e^{c+d x} \sinh (a+b x)}{9 b^4-10 b^2 d^2+d^4}+\frac {3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5585, 5583}
\begin {gather*} -\frac {d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac {3 b e^{c+d x} \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2-d^2}-\frac {6 b^2 d e^{c+d x} \cosh (a+b x)}{9 b^4-10 b^2 d^2+d^4}+\frac {6 b^3 e^{c+d x} \sinh (a+b x)}{9 b^4-10 b^2 d^2+d^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 5583
Rule 5585
Rubi steps
\begin {align*} \int e^{c+d x} \cosh ^3(a+b x) \, dx &=-\frac {d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac {3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}+\frac {\left (6 b^2\right ) \int e^{c+d x} \cosh (a+b x) \, dx}{9 b^2-d^2}\\ &=-\frac {6 b^2 d e^{c+d x} \cosh (a+b x)}{9 b^4-10 b^2 d^2+d^4}-\frac {d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac {6 b^3 e^{c+d x} \sinh (a+b x)}{9 b^4-10 b^2 d^2+d^4}+\frac {3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.33, size = 106, normalized size = 0.74 \begin {gather*} \frac {e^{c+d x} \left (3 d \left (-9 b^2+d^2\right ) \cosh (a+b x)+\left (-b^2 d+d^3\right ) \cosh (3 (a+b x))+6 b \left (5 b^2-d^2+\left (b^2-d^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)\right )}{4 \left (9 b^4-10 b^2 d^2+d^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.19, size = 178, normalized size = 1.24
method | result | size |
default | \(\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}-\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}-\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{8 \left (3 b -d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}\) | \(178\) |
risch | \(\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}+27 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}-3 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}+3 d^{3} {\mathrm e}^{2 b x +2 a}-3 b^{3}-b^{2} d +3 d^{2} b +d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{8 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 381 vs.
\(2 (140) = 280\).
time = 0.41, size = 381, normalized size = 2.65 \begin {gather*} -\frac {3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{3} - b d^{2} + 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) - 3 \, {\left (9 \, b^{3} - b d^{2} + 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1015 vs.
\(2 (131) = 262\).
time = 2.90, size = 1015, normalized size = 7.05 \begin {gather*} \begin {cases} x e^{c} \cosh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge d = 0 \\- \frac {3 x e^{c} e^{d x} \sinh ^{3}{\left (a - d x \right )}}{8} - \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a - d x \right )} \cosh {\left (a - d x \right )}}{8} + \frac {3 x e^{c} e^{d x} \sinh {\left (a - d x \right )} \cosh ^{2}{\left (a - d x \right )}}{8} + \frac {3 x e^{c} e^{d x} \cosh ^{3}{\left (a - d x \right )}}{8} + \frac {5 e^{c} e^{d x} \sinh ^{3}{\left (a - d x \right )}}{8 d} + \frac {e^{c} e^{d x} \sinh ^{2}{\left (a - d x \right )} \cosh {\left (a - d x \right )}}{4 d} - \frac {e^{c} e^{d x} \sinh {\left (a - d x \right )} \cosh ^{2}{\left (a - d x \right )}}{d} - \frac {3 e^{c} e^{d x} \cosh ^{3}{\left (a - d x \right )}}{8 d} & \text {for}\: b = - d \\\frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{3} \right )}}{8} + \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{3} \right )} \cosh {\left (a - \frac {d x}{3} \right )}}{8} + \frac {3 x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{3} \right )} \cosh ^{2}{\left (a - \frac {d x}{3} \right )}}{8} + \frac {x e^{c} e^{d x} \cosh ^{3}{\left (a - \frac {d x}{3} \right )}}{8} - \frac {e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{3} \right )}}{8 d} + \frac {3 e^{c} e^{d x} \sinh {\left (a - \frac {d x}{3} \right )} \cosh ^{2}{\left (a - \frac {d x}{3} \right )}}{4 d} + \frac {9 e^{c} e^{d x} \cosh ^{3}{\left (a - \frac {d x}{3} \right )}}{8 d} & \text {for}\: b = - \frac {d}{3} \\- \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{3} \right )}}{8} + \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{3} \right )} \cosh {\left (a + \frac {d x}{3} \right )}}{8} - \frac {3 x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{3} \right )} \cosh ^{2}{\left (a + \frac {d x}{3} \right )}}{8} + \frac {x e^{c} e^{d x} \cosh ^{3}{\left (a + \frac {d x}{3} \right )}}{8} + \frac {e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{3} \right )}}{8 d} - \frac {3 e^{c} e^{d x} \sinh {\left (a + \frac {d x}{3} \right )} \cosh ^{2}{\left (a + \frac {d x}{3} \right )}}{4 d} + \frac {9 e^{c} e^{d x} \cosh ^{3}{\left (a + \frac {d x}{3} \right )}}{8 d} & \text {for}\: b = \frac {d}{3} \\\frac {3 x e^{c} e^{d x} \sinh ^{3}{\left (a + d x \right )}}{8} - \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a + d x \right )} \cosh {\left (a + d x \right )}}{8} - \frac {3 x e^{c} e^{d x} \sinh {\left (a + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{8} + \frac {3 x e^{c} e^{d x} \cosh ^{3}{\left (a + d x \right )}}{8} - \frac {3 e^{c} e^{d x} \sinh ^{3}{\left (a + d x \right )}}{8 d} + \frac {3 e^{c} e^{d x} \sinh {\left (a + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{4 d} - \frac {e^{c} e^{d x} \cosh ^{3}{\left (a + d x \right )}}{8 d} & \text {for}\: b = d \\- \frac {6 b^{3} e^{c} e^{d x} \sinh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {9 b^{3} e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {6 b^{2} d e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {7 b^{2} d e^{c} e^{d x} \cosh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {3 b d^{2} e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {d^{3} e^{c} e^{d x} \cosh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 86, normalized size = 0.60 \begin {gather*} \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \, {\left (3 \, b + d\right )}} + \frac {3 \, e^{\left (b x + d x + a + c\right )}}{8 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{8 \, {\left (b - d\right )}} - \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{8 \, {\left (3 \, b - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.20, size = 125, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (9\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )-6\,b^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3-7\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^3+6\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2-3\,b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+d^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\right )}{9\,b^4-10\,b^2\,d^2+d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________