Optimal. Leaf size=53 \[ \frac {2 x \tanh \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}}{d}-\frac {4 \sqrt {a \cosh (c+d x)+a}}{d^2} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3319, 3296, 2638} \[ \frac {2 x \tanh \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}}{d}-\frac {4 \sqrt {a \cosh (c+d x)+a}}{d^2} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x \sqrt {a+a \cosh (c+d x)} \, dx &=\left (\sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int x \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \, dx\\ &=\frac {2 x \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {\left (2 \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \sinh \left (\frac {c}{2}+\frac {d x}{2}\right ) \, dx}{d}\\ &=-\frac {4 \sqrt {a+a \cosh (c+d x)}}{d^2}+\frac {2 x \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 34, normalized size = 0.64 \[ \frac {2 \left (d x \tanh \left (\frac {1}{2} (c+d x)\right )-2\right ) \sqrt {a (\cosh (c+d x)+1)}}{d^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 67, normalized size = 1.26 \[ \frac {\sqrt {2} {\left (\sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 2 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 2 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 64, normalized size = 1.21 \[ \frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{d x +c}+1\right )^{2} {\mathrm e}^{-d x -c}}\, \left (d x \,{\mathrm e}^{d x +c}-d x -2 \,{\mathrm e}^{d x +c}-2\right )}{\left ({\mathrm e}^{d x +c}+1\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 60, normalized size = 1.13 \[ -\frac {{\left (\sqrt {2} \sqrt {a} d x - {\left (\sqrt {2} \sqrt {a} d x e^{c} - 2 \, \sqrt {2} \sqrt {a} e^{c}\right )} e^{\left (d x\right )} + 2 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 56, normalized size = 1.06 \[ \frac {2\,x\,\mathrm {sinh}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{d\,\mathrm {cosh}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {4\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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