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 = 0.05, 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 = {2749, 2729,
2727} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rule 2749
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 \begin {gather*} \frac {2 a^2 \sec ^3(c+d x)}{3 d}+\frac {a^2 \sec ^2(c+d x) \tan (c+d x)}{d}-\frac {a^2 \tan ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 63, normalized size = 1.00
method | result | size |
risch | \(-\frac {2 i a^{2} \left (-i+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}\) | \(38\) |
derivativedivides | \(\frac {\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {2 a^{2}}{3 \cos \left (d x +c \right )^{3}}-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(63\) |
default | \(\frac {\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {2 a^{2}}{3 \cos \left (d x +c \right )^{3}}-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(63\) |
norman | \(\frac {-\frac {4 a^{2}}{3 d}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {16 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {20 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 52, normalized size = 0.83 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 97, normalized size = 1.54 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.87, size = 54, normalized size = 0.86 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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