Optimal. Leaf size=20 \[ \frac {a^3}{d (a-a \sin (c+d x))} \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac {a^3}{d (a-a \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 32, normalized size = 1.60 \[ \frac {a^2}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 19, normalized size = 0.95 \[ -\frac {a^{2}}{d \sin \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 30, normalized size = 1.50 \[ \frac {2 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 75, normalized size = 3.75 \[ \frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sin \left (d x +c \right )}{2 d}+\frac {a^{2}}{d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 18, normalized size = 0.90 \[ -\frac {a^{2}}{d {\left (\sin \left (d x + c\right ) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.90 \[ -\frac {a^2}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]
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 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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