Optimal. Leaf size=127 \[ \frac {b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5620, 5582}
\begin {gather*} -\frac {d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 5582
Rule 5620
Rubi steps
\begin {align*} \int e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\int \left (\frac {1}{4} e^{c+d x} \sinh (a+b x)+\frac {1}{4} e^{c+d x} \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac {1}{4} \int e^{c+d x} \sinh (a+b x) \, dx+\frac {1}{4} \int e^{c+d x} \sinh (3 a+3 b x) \, dx\\ &=\frac {b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 80, normalized size = 0.63 \begin {gather*} \frac {1}{4} e^{c+d x} \left (\frac {b \cosh (a+b x)-d \sinh (a+b x)}{(b-d) (b+d)}+\frac {3 b \cosh (3 (a+b x))-d \sinh (3 (a+b x))}{9 b^2-d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.07, size = 178, normalized size = 1.40
method | result | size |
default | \(-\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{8 b +8 d}-\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{8 \left (3 b -d \right )}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}+\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{8 b -8 d}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{8 b +8 d}+\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}\) | \(178\) |
risch | \(\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}+9 b^{3} {\mathrm e}^{4 b x +4 a}-9 b^{2} d \,{\mathrm e}^{4 b x +4 a}-b \,d^{2} {\mathrm e}^{4 b x +4 a}+d^{3} {\mathrm e}^{4 b x +4 a}+9 b^{3} {\mathrm e}^{2 b x +2 a}+9 b^{2} d \,{\mathrm e}^{2 b x +2 a}-b \,d^{2} {\mathrm e}^{2 b x +2 a}-d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+b^{2} d -3 d^{2} b -d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{8 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs.
\(2 (115) = 230\).
time = 0.37, size = 381, normalized size = 3.00 \begin {gather*} \frac {9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - {\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - {\left (9 \, b^{2} d - d^{3} + 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left (3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} + {\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} + {\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) - {\left (9 \, b^{2} d - d^{3} + 3 \, {\left (b^{2} d - d^{3}\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.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 971 vs.
\(2 (109) = 218\).
time = 2.83, size = 971, normalized size = 7.65 \begin {gather*} \begin {cases} x e^{c} \sinh {\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {for}\: b = 0 \wedge d = 0 \\- \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - d x \right )}}{8} - \frac {x e^{c} e^{d x} \sinh ^{2}{\left (a - d x \right )} \cosh {\left (a - d x \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a - d x \right )} \cosh ^{2}{\left (a - d x \right )}}{8} + \frac {x e^{c} e^{d x} \cosh ^{3}{\left (a - d x \right )}}{8} - \frac {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} \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 {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 {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 + d x \right )}}{8} + \frac {x e^{c} e^{d x} \sinh ^{2}{\left (a + d x \right )} \cosh {\left (a + d x \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{8} - \frac {x e^{c} e^{d x} \cosh ^{3}{\left (a + d x \right )}}{8} + \frac {e^{c} e^{d x} \sinh ^{3}{\left (a + d x \right )}}{8 d} - \frac {e^{c} e^{d x} \sinh {\left (a + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{4 d} + \frac {3 e^{c} e^{d x} \cosh ^{3}{\left (a + d x \right )}}{8 d} & \text {for}\: b = d \\\frac {3 b^{3} e^{c} e^{d x} \cosh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {2 b^{2} d e^{c} e^{d x} \sinh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {3 b^{2} d 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 {2 b d^{2} 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 {b d^{2} e^{c} e^{d x} \cosh ^{3}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {d^{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}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 86, normalized size = 0.68 \begin {gather*} \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (b x + d x + a + c\right )}}{8 \, {\left (b + d\right )}} + \frac {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.
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Mupad [B]
time = 2.18, size = 126, normalized size = 0.99 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3-3\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+2\,b^2\,d\,{\mathrm {sinh}\left (a+b\,x\right )}^3-b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3-2\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2+d^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\right )}{9\,b^4-10\,b^2\,d^2+d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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