Optimal. Leaf size=54 \[ -\frac {2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d} \]
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Rubi [A]
time = 0.03, 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 = {5419, 5413,
3377, 2718} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5413
Rule 5419
Rubi steps
\begin {align*} \int \cosh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int \cosh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \text {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 \text {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 = 0.05, size = 50, normalized size = 0.93 \begin {gather*} \frac {2 \left (-\cosh \left (a+b \sqrt {c+d x}\right )+b \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.39, size = 63, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-2 \cosh \left (a +b \sqrt {d x +c}\right )-2 a \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2} d}\) | \(63\) |
default | \(\frac {2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-2 \cosh \left (a +b \sqrt {d x +c}\right )-2 a \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2} d}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (48) = 96\).
time = 0.26, size = 110, normalized size = 2.04 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 44, normalized size = 0.81 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 65, normalized size = 1.20 \begin {gather*} \begin {cases} x \cosh {\left (a \right )} & \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 65, normalized size = 1.20 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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