Optimal. Leaf size=54 \[ \frac {2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d}-\frac {2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5311, 5305, 3296, 2638} \[ \frac {2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d}-\frac {2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5305
Rule 5311
Rubi steps
\begin {align*} \int \cosh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \cosh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d}-\frac {2 \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d}\\ &=-\frac {2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 50, normalized size = 0.93 \[ \frac {2 \left (b \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )-\cosh \left (a+b \sqrt {c+d x}\right )\right )}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 44, normalized size = 0.81 \[ \frac {2 \, {\left (\sqrt {d x + c} b \sinh \left (\sqrt {d x + c} b + a\right ) - \cosh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 65, normalized size = 1.20 \[ \frac {{\left (\sqrt {d x + c} b - 1\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{2} d} - \frac {{\left (\sqrt {d x + c} b + 1\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 63, normalized size = 1.17 \[ \frac {2 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-2 \cosh \left (a +b \sqrt {d x +c}\right )-2 a \sinh \left (a +b \sqrt {d x +c}\right )}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 110, normalized size = 2.04 \[ -\frac {b {\left (\frac {{\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} + \frac {{\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}}\right )} - 2 \, {\left (d x + c\right )} \cosh \left (\sqrt {d x + c} b + a\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 43, normalized size = 0.80 \[ -\frac {2\,\left (\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right )-b\,\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right )\,\sqrt {c+d\,x}\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 65, normalized size = 1.20 \[ \begin {cases} x \cosh {\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cosh {\left (a + b \sqrt {c} \right )} & \text {for}\: d = 0 \\\frac {2 \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {2 \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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