Optimal. Leaf size=54 \[ \frac {2 \sin \left (a+b \sqrt {c+d x}\right )}{b^2 d}-\frac {2 \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{b d} \]
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Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3361, 3296, 2637} \[ \frac {2 \sin \left (a+b \sqrt {c+d x}\right )}{b^2 d}-\frac {2 \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3361
Rubi steps
\begin {align*} \int \sin \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=-\frac {2 \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{b d}+\frac {2 \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d}\\ &=-\frac {2 \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{b d}+\frac {2 \sin \left (a+b \sqrt {c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 50, normalized size = 0.93 \[ \frac {2 \sin \left (a+b \sqrt {c+d x}\right )-2 b \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 44, normalized size = 0.81 \[ -\frac {2 \, {\left (\sqrt {d x + c} b \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 44, normalized size = 0.81 \[ -\frac {2 \, {\left (\sqrt {d x + c} b \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 61, normalized size = 1.13 \[ \frac {2 \sin \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cos \left (a +b \sqrt {d x +c}\right )+2 a \cos \left (a +b \sqrt {d x +c}\right )}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 62, normalized size = 1.15 \[ -\frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - a \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 43, normalized size = 0.80 \[ \frac {2\,\left (\sin \left (a+b\,\sqrt {c+d\,x}\right )-b\,\cos \left (a+b\,\sqrt {c+d\,x}\right )\,\sqrt {c+d\,x}\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 66, normalized size = 1.22 \[ \begin {cases} x \sin {\relax (a )} & \text {for}\: b = 0 \wedge d = 0 \\x \sin {\left (a + b \sqrt {c} \right )} & \text {for}\: d = 0 \\x \sin {\relax (a )} & \text {for}\: b = 0 \\- \frac {2 \sqrt {c + d x} \cos {\left (a + b \sqrt {c + d x} \right )}}{b d} + \frac {2 \sin {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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