Optimal. Leaf size=55 \[ -\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {c x}{2}+\frac {d x^2}{4} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3310} \[ -\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {c x}{2}+\frac {d x^2}{4} \]
Antiderivative was successfully verified.
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Rule 3310
Rubi steps
\begin {align*} \int (c+d x) \cosh ^2(a+b x) \, dx &=-\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x) \, dx\\ &=\frac {c x}{2}+\frac {d x^2}{4}-\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 51, normalized size = 0.93 \[ \frac {2 b ((c+d x) \sinh (2 (a+b x))+2 a c+b x (2 c+d x))-d \cosh (2 (a+b x))}{8 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 66, normalized size = 1.20 \[ \frac {2 \, b^{2} d x^{2} + 4 \, b^{2} c x - d \cosh \left (b x + a\right )^{2} + 4 \, {\left (b d x + b c\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - d \sinh \left (b x + a\right )^{2}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 63, normalized size = 1.15 \[ \frac {1}{4} \, d x^{2} + \frac {1}{2} \, c x + \frac {{\left (2 \, b d x + 2 \, b c - d\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{2}} - \frac {{\left (2 \, b d x + 2 \, b c + d\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 103, normalized size = 1.87 \[ \frac {\frac {d \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b}-\frac {d a \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}+c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 88, normalized size = 1.60 \[ \frac {1}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} d + \frac {1}{8} \, c {\left (4 \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 58, normalized size = 1.05 \[ \frac {b^2\,d\,x^2-\frac {d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}+b\,c\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+2\,b^2\,c\,x+b\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 126, normalized size = 2.29 \[ \begin {cases} - \frac {c x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac {d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {c \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {d \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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