Optimal. Leaf size=38 \[ \frac {a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan (c+d x)}{d}+a^2 x \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4120, 3473, 8} \[ \frac {a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan (c+d x)}{d}+a^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 4120
Rubi steps
\begin {align*} \int \left (a-a \sec ^2(c+d x)\right )^2 \, dx &=a^2 \int \tan ^4(c+d x) \, dx\\ &=\frac {a^2 \tan ^3(c+d x)}{3 d}-a^2 \int \tan ^2(c+d x) \, dx\\ &=-\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^3(c+d x)}{3 d}+a^2 \int 1 \, dx\\ &=a^2 x-\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.11 \[ a^2 \left (\frac {\tan ^{-1}(\tan (c+d x))}{d}+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 56, normalized size = 1.47 \[ \frac {3 \, a^{2} d x \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 39, normalized size = 1.03 \[ \frac {a^{2} \tan \left (d x + c\right )^{3} + 3 \, {\left (d x + c\right )} a^{2} - 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.02, size = 49, normalized size = 1.29 \[ \frac {-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-2 a^{2} \tan \left (d x +c \right )+a^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 45, normalized size = 1.18 \[ a^{2} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2}}{3 \, d} - \frac {2 \, a^{2} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 33, normalized size = 0.87 \[ a^2\,x-\frac {a^2\,\left (3\,\mathrm {tan}\left (c+d\,x\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\right )}{3\,d} \]
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 \[ a^{2} \left (\int 1\, dx + \int \left (- 2 \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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