Optimal. Leaf size=18 \[ \frac {2 d \sqrt {d \sec (a+b x)}}{b} \]
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Rubi [A]
time = 0.03, 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 \sqrt {d \sec (a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2702
Rubi steps
\begin {align*} \int (d \sec (a+b x))^{3/2} \sin (a+b x) \, dx &=\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac {2 d \sqrt {d \sec (a+b x)}}{b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 d \sqrt {d \sec (a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {2 d \sqrt {d \sec \left (b x +a \right )}}{b}\) | \(17\) |
default | \(\frac {2 d \sqrt {d \sec \left (b x +a \right )}}{b}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 23, normalized size = 1.28 \begin {gather*} \frac {2 \, \left (\frac {d}{\cos \left (b x + a\right )}\right )^{\frac {3}{2}} \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 = 3.35, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \, d \sqrt {\frac {d}{\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 \left (d \sec {\left (a + b x \right )}\right )^{\frac {3}{2}} \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 = 0.43, size = 25, normalized size = 1.39 \begin {gather*} \frac {2 \, d^{2} \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{\sqrt {d \cos \left (b x + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 18, normalized size = 1.00 \begin {gather*} \frac {2\,d\,\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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