Optimal. Leaf size=139 \[ -\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac {3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 139, 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 = {5584, 5582}
\begin {gather*} \frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}-\frac {3 d e^{a+b x} \sinh ^2(c+d x) \cosh (c+d x)}{b^2-9 d^2}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 5582
Rule 5584
Rubi steps
\begin {align*} \int e^{a+b x} \sinh ^3(c+d x) \, dx &=-\frac {3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac {\left (6 d^2\right ) \int e^{a+b x} \sinh (c+d x) \, dx}{b^2-9 d^2}\\ &=-\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac {3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 108, normalized size = 0.78 \begin {gather*} \frac {e^{a+b x} \left (3 d \left (b^2-9 d^2\right ) \cosh (c+d x)+\left (-3 b^2 d+3 d^3\right ) \cosh (3 (c+d x))+2 b \left (-b^2+13 d^2+\left (b^2-d^2\right ) \cosh (2 (c+d x))\right ) \sinh (c+d x)\right )}{4 \left (b^4-10 b^2 d^2+9 d^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.82, size = 166, normalized size = 1.19
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +3 d x +a +3 c}}{8 b +24 d}-\frac {3 \,{\mathrm e}^{b x +d x +a +c}}{8 \left (b +d \right )}+\frac {3 \,{\mathrm e}^{b x -d x +a -c}}{8 \left (b -d \right )}-\frac {{\mathrm e}^{b x -3 d x +a -3 c}}{8 \left (b -3 d \right )}\) | \(85\) |
default | \(-\frac {\sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}-\frac {\cosh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}\) | \(166\) |
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. 316 vs.
\(2 (135) = 270\).
time = 0.33, size = 316, normalized size = 2.27 \begin {gather*} -\frac {3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} - {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + 9 \, {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + {\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right ) - {\left (b^{3} - 9 \, b d^{2} - {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{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. 1085 vs.
\(2 (131) = 262\).
time = 3.44, size = 1085, normalized size = 7.81 \begin {gather*} \begin {cases} x e^{a} \sinh ^{3}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{a} e^{- 3 d x} \sinh ^{3}{\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{- 3 d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{- 3 d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {x e^{a} e^{- 3 d x} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {e^{a} e^{- 3 d x} \sinh ^{3}{\left (c + d x \right )}}{24 d} + \frac {e^{a} e^{- 3 d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {5 e^{a} e^{- 3 d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} + \frac {11 e^{a} e^{- 3 d x} \cosh ^{3}{\left (c + d x \right )}}{24 d} & \text {for}\: b = - 3 d \\\frac {3 x e^{a} e^{- d x} \sinh ^{3}{\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{- d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x e^{a} e^{- d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {3 x e^{a} e^{- d x} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {3 e^{a} e^{- d x} \sinh ^{3}{\left (c + d x \right )}}{8 d} + \frac {e^{a} e^{- d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {e^{a} e^{- d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} - \frac {5 e^{a} e^{- d x} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: b = - d \\\frac {3 x e^{a} e^{d x} \sinh ^{3}{\left (c + d x \right )}}{8} - \frac {3 x e^{a} e^{d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x e^{a} e^{d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{d x} \cosh ^{3}{\left (c + d x \right )}}{8} - \frac {3 e^{a} e^{d x} \sinh ^{3}{\left (c + d x \right )}}{8 d} + \frac {e^{a} e^{d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {e^{a} e^{d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} - \frac {5 e^{a} e^{d x} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: b = d \\\frac {x e^{a} e^{3 d x} \sinh ^{3}{\left (c + d x \right )}}{8} - \frac {3 x e^{a} e^{3 d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{3 d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {x e^{a} e^{3 d x} \cosh ^{3}{\left (c + d x \right )}}{8} - \frac {e^{a} e^{3 d x} \sinh ^{3}{\left (c + d x \right )}}{24 d} + \frac {e^{a} e^{3 d x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {5 e^{a} e^{3 d x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} + \frac {11 e^{a} e^{3 d x} \cosh ^{3}{\left (c + d x \right )}}{24 d} & \text {for}\: b = 3 d \\\frac {b^{3} e^{a} e^{b x} \sinh ^{3}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {3 b^{2} d e^{a} e^{b x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {7 b d^{2} e^{a} e^{b x} \sinh ^{3}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} + \frac {6 b d^{2} e^{a} e^{b x} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} + \frac {9 d^{3} e^{a} e^{b x} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {6 d^{3} e^{a} e^{b x} \cosh ^{3}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 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 = 0.41, size = 84, normalized size = 0.60 \begin {gather*} \frac {e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{8 \, {\left (b + 3 \, 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 (b x - 3 \, d x + a - 3 \, c\right )}}{8 \, {\left (b - 3 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.32, size = 127, normalized size = 0.91 \begin {gather*} -\frac {{\mathrm {e}}^{a+b\,x}\,\left (-b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3+3\,b^2\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2-6\,b\,d^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )+7\,b\,d^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3+6\,d^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3-9\,d^3\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\right )}{b^4-10\,b^2\,d^2+9\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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