Optimal. Leaf size=50 \[ -3 a^3 x+\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2749, 2759,
2761, 8} \begin {gather*} \frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac {3 a^3 \cos (c+d x)}{d}-3 a^3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2749
Rule 2759
Rule 2761
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^6 \int \frac {\cos ^4(c+d x)}{(a-a \sin (c+d x))^3} \, dx\\ &=\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-\left (3 a^4\right ) \int \frac {\cos ^2(c+d x)}{a-a \sin (c+d x)} \, dx\\ &=\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-\left (3 a^3\right ) \int 1 \, dx\\ &=-3 a^3 x+\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 55, normalized size = 1.10 \begin {gather*} \frac {4 \sqrt {2} a^3 \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sec (c+d x) \sqrt {1+\sin (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 87, normalized size = 1.74
method | result | size |
risch | \(-3 a^{3} x +\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {8 a^{3}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\) | \(64\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {3 a^{3}}{\cos \left (d x +c \right )}+a^{3} \tan \left (d x +c \right )}{d}\) | \(87\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {3 a^{3}}{\cos \left (d x +c \right )}+a^{3} \tan \left (d x +c \right )}{d}\) | \(87\) |
norman | \(\frac {3 a^{3} x -\frac {10 a^{3}}{d}-\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {24 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {22 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+6 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {6 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {26 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(229\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 68, normalized size = 1.36 \begin {gather*} -\frac {3 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - a^{3} \tan \left (d x + c\right ) - \frac {3 \, a^{3}}{\cos \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 101, normalized size = 2.02 \begin {gather*} -\frac {3 \, a^{3} d x - a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (3 \, a^{3} d x - 5 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{3} d x - a^{3} \cos \left (d x + c\right ) + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \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^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\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 = 5.61, size = 91, normalized size = 1.82 \begin {gather*} -\frac {3 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.78, size = 138, normalized size = 2.76 \begin {gather*} -3\,a^3\,x-\frac {3\,a^3\,\left (c+d\,x\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^3\,\left (c+d\,x\right )-a^3\,\left (3\,c+3\,d\,x-2\right )\right )-a^3\,\left (3\,c+3\,d\,x-10\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a^3\,\left (c+d\,x\right )-a^3\,\left (3\,c+3\,d\,x-8\right )\right )}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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