Optimal. Leaf size=38 \[ \frac {2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 x \]
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Rubi [A] time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2670, 2680, 8} \[ \frac {2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2670
Rule 2680
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac {\cos ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 \int 1 \, dx\\ &=-a^2 x+\frac {2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 1.97 \[ \frac {2 a^2 \sqrt {\sin (c+d x)+1} \left (\sqrt {1-\sin (c+d x)} \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )+\sqrt {\sin (c+d x)+1}\right ) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 74, normalized size = 1.95 \[ -\frac {a^{2} d x - 2 \, a^{2} + {\left (a^{2} d x - 2 \, a^{2}\right )} \cos \left (d x + c\right ) - {\left (a^{2} d x + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 33, normalized size = 0.87 \[ -\frac {{\left (d x + c\right )} a^{2} + \frac {4 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 47, normalized size = 1.24 \[ \frac {a^{2} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {2 a^{2}}{\cos \left (d x +c \right )}+a^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 47, normalized size = 1.24 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - a^{2} \tan \left (d x + c\right ) - \frac {2 \, a^{2}}{\cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 28, normalized size = 0.74 \[ -a^2\,x-\frac {4\,a^2}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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