Optimal. Leaf size=57 \[ -\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 272,
45} \begin {gather*} -\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{8 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^3} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^3}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{16 b}\\ &=\frac {\text {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{16 b}\\ &=-\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 47, normalized size = 0.82 \begin {gather*} \frac {-e^{-2 (a+b x)}+3 e^{2 (a+b x)}+\frac {1}{2} e^{4 (a+b x)}+6 b x}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.14, size = 61, normalized size = 1.07
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 b x -2 a}}{16 b}+\frac {3 \,{\mathrm e}^{2 b x +2 a}}{16 b}+\frac {{\mathrm e}^{4 b x +4 a}}{32 b}+\frac {3 x}{8}\) | \(47\) |
default | \(\frac {3 x}{8}+\frac {\sinh \left (2 b x +2 a \right )}{4 b}+\frac {\sinh \left (4 b x +4 a \right )}{32 b}+\frac {\cosh \left (2 b x +2 a \right )}{8 b}+\frac {\cosh \left (4 b x +4 a \right )}{32 b}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 53, normalized size = 0.93 \begin {gather*} \frac {3 \, {\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{32 \, b} + \frac {3 \, e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, 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. 95 vs.
\(2 (46) = 92\).
time = 0.36, size = 95, normalized size = 1.67 \begin {gather*} -\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, \sinh \left (b x + a\right )^{3} - 6 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + a\right ) + 3 \, {\left (4 \, b x - 3 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )}{32 \, {\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. 182 vs.
\(2 (48) = 96\).
time = 1.24, size = 182, normalized size = 3.19 \begin {gather*} \begin {cases} \frac {3 x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {3 x e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8} - \frac {3 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x e^{a} \cosh ^{3}{\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 = 57, normalized size = 1.00 \begin {gather*} \frac {12 \, b x - 2 \, {\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 12 \, a + e^{\left (4 \, b x + 4 \, a\right )} + 6 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 42, normalized size = 0.74 \begin {gather*} \frac {3\,x}{8}+\frac {\frac {3\,{\mathrm {e}}^{2\,a+2\,b\,x}}{16}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{16}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{32}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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