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 = 0.08, 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 = {2282, 12, 446, 88} \[ -\frac {e^{-4 a-4 b x}}{128 b}-\frac {e^{-2 a-2 b x}}{64 b}-\frac {e^{2 a+2 b x}}{32 b}+\frac {e^{4 a+4 b x}}{128 b}+\frac {e^{6 a+6 b x}}{192 b}-\frac {x}{16} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (1+x^2\right )^3}{32 x^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (1+x^2\right )^3}{x^5} \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2 (1+x)^3}{x^3} \, dx,x,e^{2 a+2 b x}\right )}{64 b}\\ &=\frac {\operatorname {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 = 0.11, size = 67, normalized size = 0.74 \[ -\frac {3 e^{-4 (a+b x)}+6 e^{-2 (a+b x)}+12 e^{2 (a+b x)}-3 e^{4 (a+b x)}-2 e^{6 (a+b x)}+24 b x}{384 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 165, normalized size = 1.81 \[ -\frac {\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 5 \, \sinh \left (b x + a\right )^{5} - {\left (50 \, \cosh \left (b x + a\right )^{2} + 9\right )} \sinh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{3} + {\left (10 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 12 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + a\right ) - {\left (25 \, \cosh \left (b x + a\right )^{4} + 24 \, b x + 27 \, \cosh \left (b x + a\right )^{2} - 12\right )} \sinh \left (b x + a\right )}{384 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 81, normalized size = 0.89 \[ -\frac {24 \, b x - 3 \, {\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 24 \, a - 2 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 84, normalized size = 0.92 \[ \frac {\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}+\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{6}-\frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{6}-\frac {b x}{16}-\frac {a}{16}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 77, normalized size = 0.85 \[ -\frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {b x + a}{16 \, b} + \frac {2 \, e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 65, normalized size = 0.71 \[ -\frac {6\,{\mathrm {e}}^{-2\,a-2\,b\,x}+12\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{-4\,a-4\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{6\,a+6\,b\,x}+24\,b\,x}{384\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 170.93, size = 325, normalized size = 3.57 \[ \begin {cases} \frac {x e^{a} e^{b x} \sinh ^{5}{\left (a + b x \right )}}{16} - \frac {x e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16} - \frac {x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8} + \frac {x e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac {x e^{a} e^{b x} \cosh ^{5}{\left (a + b x \right )}}{16} - \frac {13 e^{a} e^{b x} \sinh ^{5}{\left (a + b x \right )}}{96 b} + \frac {7 e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{96 b} + \frac {e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} - \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{96 b} + \frac {5 e^{a} e^{b x} \cosh ^{5}{\left (a + b x \right )}}{96 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\relax (a )} \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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