Optimal. Leaf size=91 \[ \frac {e^{-4 a-4 b x}}{128 b}-\frac {e^{-2 a-2 b x}}{64 b}-\frac {e^{2 a+2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{128 b}+\frac {e^{6 a+6 b x}}{192 b}+\frac {x}{16} \]
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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 12, 457,
90} \begin {gather*} \frac {e^{-4 a-4 b x}}{128 b}-\frac {e^{-2 a-2 b x}}{64 b}-\frac {e^{2 a+2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{128 b}+\frac {e^{6 a+6 b x}}{192 b}+\frac {x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 457
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{32 x^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{x^5} \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac {\text {Subst}\left (\int \frac {(-1+x)^3 (1+x)^2}{x^3} \, dx,x,e^{2 a+2 b x}\right )}{64 b}\\ &=\frac {\text {Subst}\left (\int \left (-2-\frac {1}{x^3}+\frac {1}{x^2}+\frac {2}{x}-x+x^2\right ) \, dx,x,e^{2 a+2 b x}\right )}{64 b}\\ &=\frac {e^{-4 a-4 b x}}{128 b}-\frac {e^{-2 a-2 b x}}{64 b}-\frac {e^{2 a+2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{128 b}+\frac {e^{6 a+6 b x}}{192 b}+\frac {x}{16}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 67, normalized size = 0.74 \begin {gather*} \frac {3 e^{-4 (a+b x)}-6 e^{-2 (a+b x)}-12 e^{2 (a+b x)}-3 e^{4 (a+b x)}+2 e^{6 (a+b x)}+24 b x}{384 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 89, normalized size = 0.98
method | result | size |
risch | \(\frac {{\mathrm e}^{-4 b x -4 a}}{128 b}-\frac {{\mathrm e}^{-2 b x -2 a}}{64 b}-\frac {{\mathrm e}^{2 b x +2 a}}{32 b}-\frac {{\mathrm e}^{4 b x +4 a}}{128 b}+\frac {{\mathrm e}^{6 b x +6 a}}{192 b}+\frac {x}{16}\) | \(75\) |
derivativedivides | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{8}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}+\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}}{b}\) | \(89\) |
default | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{8}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}+\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}}{b}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 77, normalized size = 0.85 \begin {gather*} -\frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} + \frac {b x + a}{16 \, b} + \frac {2 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \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. 167 vs.
\(2 (74) = 148\).
time = 0.40, size = 167, normalized size = 1.84 \begin {gather*} \frac {5 \, \cosh \left (b x + a\right )^{5} + 25 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - \sinh \left (b x + a\right )^{5} - {\left (10 \, \cosh \left (b x + a\right )^{2} - 3\right )} \sinh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )^{3} + {\left (50 \, \cosh \left (b x + a\right )^{3} - 27 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 12 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right ) - {\left (5 \, \cosh \left (b x + a\right )^{4} + 24 \, b x - 9 \, \cosh \left (b x + a\right )^{2} + 12\right )} \sinh \left (b x + a\right )}{384 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\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. 325 vs.
\(2 (73) = 146\).
time = 7.21, size = 325, normalized size = 3.57 \begin {gather*} \begin {cases} - \frac {x e^{a} e^{b x} \sinh ^{5}{\left (a + b x \right )}}{16} + \frac {x e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16} + \frac {x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} - \frac {x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8} - \frac {x e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac {x e^{a} e^{b x} \cosh ^{5}{\left (a + b x \right )}}{16} + \frac {5 e^{a} e^{b x} \sinh ^{5}{\left (a + b x \right )}}{96 b} + \frac {e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{96 b} - \frac {e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{6 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {7 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{96 b} - \frac {13 e^{a} e^{b x} \cosh ^{5}{\left (a + b x \right )}}{96 b} & \text {for}\: b \neq 0 \\x e^{a} \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.
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Giac [A]
time = 0.40, size = 81, normalized size = 0.89 \begin {gather*} \frac {24 \, b x - 3 \, {\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 24 \, a + 2 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 65, normalized size = 0.71 \begin {gather*} -\frac {6\,{\mathrm {e}}^{-2\,a-2\,b\,x}+12\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{-4\,a-4\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{6\,a+6\,b\,x}-24\,b\,x}{384\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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