Optimal. Leaf size=18 \[ -\frac {2 d}{b \sqrt {d \sec (a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2702, 30}
\begin {gather*} -\frac {2 d}{b \sqrt {d \sec (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2702
Rubi steps
\begin {align*} \int \sqrt {d \sec (a+b x)} \sin (a+b x) \, dx &=\frac {d \text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=-\frac {2 d}{b \sqrt {d \sec (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 d}{b \sqrt {d \sec (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(-\frac {2 d}{b \sqrt {d \sec \left (b x +a \right )}}\) | \(17\) |
default | \(-\frac {2 d}{b \sqrt {d \sec \left (b x +a \right )}}\) | \(17\) |
risch | \(-\frac {2 \sqrt {2}\, \sqrt {\frac {d \,{\mathrm e}^{i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}+1}}\, \cos \left (b x +a \right )}{b}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, \sqrt {\frac {d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.22, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, \sqrt {\frac {d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end {gather*}
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 \begin {gather*} \int \sqrt {d \sec {\left (a + b x \right )}} \sin {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 22, normalized size = 1.22 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2\,\cos \left (a+b\,x\right )\,\sqrt {\frac {d}{\cos \left (a+b\,x\right )}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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