Optimal. Leaf size=20 \[ \frac {2 d (d \sec (a+b x))^{5/2}}{5 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 20, 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 (d \sec (a+b x))^{5/2}}{5 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))^{7/2} \sin (a+b x) \, dx &=\frac {d \text {Subst}\left (\int x^{3/2} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac {2 d (d \sec (a+b x))^{5/2}}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 d (d \sec (a+b x))^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 17, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {2 d \left (d \sec \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b}\) | \(17\) |
default | \(\frac {2 d \left (d \sec \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 1.15 \begin {gather*} \frac {2 \, \left (\frac {d}{\cos \left (b x + a\right )}\right )^{\frac {7}{2}} \cos \left (b x + a\right )}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.19, size = 28, normalized size = 1.40 \begin {gather*} \frac {2 \, d^{3} \sqrt {\frac {d}{\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (16) = 32\).
time = 0.45, size = 33, normalized size = 1.65 \begin {gather*} \frac {2 \, d^{4} \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{5 \, \sqrt {d \cos \left (b x + a\right )} b \cos \left (b x + a\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.59, size = 77, normalized size = 3.85 \begin {gather*} \frac {8\,d^3\,\sqrt {\frac {d}{\cos \left (a+b\,x\right )}}\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+3\right )}{5\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )+6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )+10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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