Optimal. Leaf size=63 \[ \frac {a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {a^4 \cos (c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 63, 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, 2650, 2648} \[ \frac {a^4 \cos (c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2670
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac {1}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^3 \int \frac {1}{a-a \sin (c+d x)} \, dx\\ &=\frac {a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {a^3 \cos (c+d x)}{3 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.92 \[ -\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {2 a^2 \sec ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 97, normalized size = 1.54 \[ -\frac {a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2} - {\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.69, size = 54, normalized size = 0.86 \[ -\frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{2}\right )}}{3 \, d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 63, normalized size = 1.00 \[ \frac {\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {2 a^{2}}{3 \cos \left (d x +c \right )^{3}}-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 52, normalized size = 0.83 \[ \frac {a^{2} \tan \left (d x + c\right )^{3} + {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + \frac {2 \, a^{2}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 81, normalized size = 1.29 \[ -\frac {2\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3\right )}{3}}{d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3} \]
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 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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