Optimal. Leaf size=62 \[ \frac {\sinh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac {\sinh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac {\sinh (a+b x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5614, 2637} \[ \frac {\sinh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac {\sinh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac {\sinh (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5614
Rubi steps
\begin {align*} \int \cosh (a+b x) \cosh ^2(c+d x) \, dx &=\int \left (\frac {1}{2} \cosh (a+b x)+\frac {1}{4} \cosh (a-2 c+(b-2 d) x)+\frac {1}{4} \cosh (a+2 c+(b+2 d) x)\right ) \, dx\\ &=\frac {1}{4} \int \cosh (a-2 c+(b-2 d) x) \, dx+\frac {1}{4} \int \cosh (a+2 c+(b+2 d) x) \, dx+\frac {1}{2} \int \cosh (a+b x) \, dx\\ &=\frac {\sinh (a+b x)}{2 b}+\frac {\sinh (a-2 c+(b-2 d) x)}{4 (b-2 d)}+\frac {\sinh (a+2 c+(b+2 d) x)}{4 (b+2 d)}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 69, normalized size = 1.11 \[ \frac {1}{4} \left (\frac {\sinh (a+b x-2 c-2 d x)}{b-2 d}+\frac {\sinh (a+b x+2 c+2 d x)}{b+2 d}+\frac {2 \sinh (a) \cosh (b x)}{b}+\frac {2 \cosh (a) \sinh (b x)}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 115, normalized size = 1.85 \[ -\frac {4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - b^{2} \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} - 4 \, d^{2}\right )} \sinh \left (b x + a\right )}{2 \, {\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{3} - 4 \, b d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 120, normalized size = 1.94 \[ \frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (b x + a\right )}}{4 \, b} - \frac {e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} - \frac {e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 57, normalized size = 0.92 \[ \frac {\sinh \left (b x +a \right )}{2 b}+\frac {\sinh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\sinh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 68, normalized size = 1.10 \[ \frac {2\,d^2\,\mathrm {sinh}\left (a+b\,x\right )-b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+2\,b\,d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^2-b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.29, size = 408, normalized size = 6.58 \[ \begin {cases} x \cosh {\relax (a )} \cosh ^{2}{\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\left (- \frac {x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d}\right ) \cosh {\relax (a )} & \text {for}\: b = 0 \\\frac {x \sinh {\left (a - 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - 2 d x \right )}}{4} + \frac {x \cosh {\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {\sinh {\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a - 2 d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = - 2 d \\- \frac {x \sinh {\left (a + 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + 2 d x \right )}}{4} + \frac {x \cosh {\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {\sinh {\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a + 2 d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = 2 d \\\frac {b^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d \sinh {\left (c + d x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} \sinh {\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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