Optimal. Leaf size=137 \[ \frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 137, 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 (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 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 ^3(a+b x) \sinh (a+b x) \, dx &=\int \left (\frac {1}{4} e^{c+d x} \sinh (2 a+2 b x)+\frac {1}{8} e^{c+d x} \sinh (4 a+4 b x)\right ) \, dx\\ &=\frac {1}{8} \int e^{c+d x} \sinh (4 a+4 b x) \, dx+\frac {1}{4} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 86, normalized size = 0.63 \begin {gather*} \frac {1}{8} e^{c+d x} \left (\frac {4 b \cosh (2 (a+b x))-2 d \sinh (2 (a+b x))}{4 b^2-d^2}+\frac {4 b \cosh (4 (a+b x))-d \sinh (4 (a+b x))}{16 b^2-d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.85, size = 202, normalized size = 1.47
method | result | size |
default | \(-\frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 \left (2 b -d \right )}+\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 b +8 d}-\frac {\sinh \left (\left (4 b -d \right ) x +4 a -c \right )}{16 \left (4 b -d \right )}+\frac {\sinh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d}+\frac {\cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 b -8 d}+\frac {\cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 b +8 d}+\frac {\cosh \left (\left (4 b -d \right ) x +4 a -c \right )}{64 b -16 d}+\frac {\cosh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d}\) | \(202\) |
risch | \(\frac {\left (16 b^{3} {\mathrm e}^{8 b x +8 a}-4 b^{2} d \,{\mathrm e}^{8 b x +8 a}-4 b \,d^{2} {\mathrm e}^{8 b x +8 a}+d^{3} {\mathrm e}^{8 b x +8 a}+64 b^{3} {\mathrm e}^{6 b x +6 a}-32 b^{2} d \,{\mathrm e}^{6 b x +6 a}-4 b \,d^{2} {\mathrm e}^{6 b x +6 a}+2 d^{3} {\mathrm e}^{6 b x +6 a}+64 b^{3} {\mathrm e}^{2 b x +2 a}+32 b^{2} d \,{\mathrm e}^{2 b x +2 a}-4 b \,d^{2} {\mathrm e}^{2 b x +2 a}-2 d^{3} {\mathrm e}^{2 b x +2 a}+16 b^{3}+4 b^{2} d -4 d^{2} b -d^{3}\right ) {\mathrm e}^{-4 b x +d x -4 a +c}}{16 \left (4 b +d \right ) \left (2 b +d \right ) \left (4 b -d \right ) \left (2 b -d \right )}\) | \(244\) |
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. 501 vs.
\(2 (125) = 250\).
time = 0.33, size = 501, normalized size = 3.66 \begin {gather*} -\frac {{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} - {\left (16 \, b^{3} - b d^{2} + 6 \, {\left (4 \, 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 )^{2} + {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} + {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{4} + {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} + {\left (16 \, b^{3} - b d^{2} + 6 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (64 \, b^{4} - 20 \, 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. 1295 vs.
\(2 (114) = 228\).
time = 8.31, size = 1295, normalized size = 9.45 \begin {gather*} \begin {cases} x e^{c} \sinh {\left (a \right )} \cosh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge d = 0 \\- \frac {x e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{2} \right )}}{8} - \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{4} + \frac {x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh ^{3}{\left (a - \frac {d x}{2} \right )}}{4} + \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{2} \right )}}{8} + \frac {e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{2} \right )}}{8 d} - \frac {e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh ^{2}{\left (a - \frac {d x}{2} \right )}}{2 d} - \frac {e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh ^{3}{\left (a - \frac {d x}{2} \right )}}{3 d} - \frac {7 e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{2} \right )}}{24 d} & \text {for}\: b = - \frac {d}{2} \\\frac {x e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{4} \right )}}{16} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{4} \right )} \cosh {\left (a - \frac {d x}{4} \right )}}{4} + \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{4} \right )} \cosh ^{2}{\left (a - \frac {d x}{4} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{4} \right )} \cosh ^{3}{\left (a - \frac {d x}{4} \right )}}{4} + \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{4} \right )}}{16} - \frac {e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{4} \right )}}{6 d} - \frac {5 e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{4} \right )} \cosh {\left (a - \frac {d x}{4} \right )}}{12 d} + \frac {11 e^{c} e^{d x} \sinh {\left (a - \frac {d x}{4} \right )} \cosh ^{3}{\left (a - \frac {d x}{4} \right )}}{12 d} + \frac {e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{4} \right )}}{6 d} & \text {for}\: b = - \frac {d}{4} \\- \frac {x e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{4} \right )}}{16} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{4} \right )} \cosh {\left (a + \frac {d x}{4} \right )}}{4} - \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{4} \right )} \cosh ^{2}{\left (a + \frac {d x}{4} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{4} \right )} \cosh ^{3}{\left (a + \frac {d x}{4} \right )}}{4} - \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{4} \right )}}{16} + \frac {e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{4} \right )}}{6 d} - \frac {5 e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{4} \right )} \cosh {\left (a + \frac {d x}{4} \right )}}{12 d} + \frac {11 e^{c} e^{d x} \sinh {\left (a + \frac {d x}{4} \right )} \cosh ^{3}{\left (a + \frac {d x}{4} \right )}}{12 d} - \frac {e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{4} \right )}}{6 d} & \text {for}\: b = \frac {d}{4} \\\frac {x e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{2} \right )}}{8} - \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{4} + \frac {x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh ^{3}{\left (a + \frac {d x}{2} \right )}}{4} - \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{2} \right )}}{8} - \frac {e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{2} \right )}}{8 d} + \frac {e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh ^{2}{\left (a + \frac {d x}{2} \right )}}{2 d} - \frac {e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh ^{3}{\left (a + \frac {d x}{2} \right )}}{3 d} + \frac {7 e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{2} \right )}}{24 d} & \text {for}\: b = \frac {d}{2} \\- \frac {6 b^{3} e^{c} e^{d x} \sinh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {12 b^{3} e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {10 b^{3} e^{c} e^{d x} \cosh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {6 b^{2} d e^{c} e^{d x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {10 b^{2} d e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {3 b d^{2} e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {b d^{2} e^{c} e^{d x} \cosh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {d^{3} e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{4} - 20 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 = 93, normalized size = 0.68 \begin {gather*} \frac {e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \, {\left (4 \, b + d\right )}} + \frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \, {\left (2 \, b + d\right )}} + \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \, {\left (4 \, b - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 163, normalized size = 1.19 \begin {gather*} -\frac {b^3\,\left (6\,{\mathrm {e}}^{c+d\,x}-16\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}\right )+b^2\,d\,\left (4\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (a+b\,x\right )}^3+6\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right )\right )-b\,d^2\,\left (3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}-4\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}\right )-d^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )}{64\,b^4-20\,b^2\,d^2+d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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