Optimal. Leaf size=103 \[ \frac {3 a^2 x}{4}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3873, 2715, 8,
4129, 3092, 380} \begin {gather*} \frac {a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {3 a^2 x}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 380
Rule 2715
Rule 3092
Rule 3873
Rule 4129
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx+\int \cos ^5(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx+\int \cos ^3(c+d x) \left (a^2+a^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}+\frac {1}{4} \left (3 a^2\right ) \int 1 \, dx-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (2 a^2-a^2 x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {3 a^2 x}{4}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \left (2 a^2-3 a^2 x^2+a^2 x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {3 a^2 x}{4}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 61, normalized size = 0.59 \begin {gather*} \frac {a^2 (60 d x+110 \sin (c+d x)+40 \sin (2 (c+d x))+15 \sin (3 (c+d x))+5 \sin (4 (c+d x))+\sin (5 (c+d x)))}{80 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 96, normalized size = 0.93
method | result | size |
risch | \(\frac {3 a^{2} x}{4}+\frac {11 a^{2} \sin \left (d x +c \right )}{8 d}+\frac {a^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{16 d}+\frac {3 a^{2} \sin \left (3 d x +3 c \right )}{16 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{2 d}\) | \(90\) |
derivativedivides | \(\frac {\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(96\) |
default | \(\frac {\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(96\) |
norman | \(\frac {-\frac {3 a^{2} x}{4}-\frac {13 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {5 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {27 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {37 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {11 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {3 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-3 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {15 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+3 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 95, normalized size = 0.92 \begin {gather*} \frac {16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 76, normalized size = 0.74 \begin {gather*} \frac {15 \, a^{2} d x + {\left (4 \, a^{2} \cos \left (d x + c\right )^{4} + 10 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 15 \, a^{2} \cos \left (d x + c\right ) + 24 \, a^{2}\right )} \sin \left (d x + c\right )}{20 \, 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^{2} \left (\int 2 \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cos ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{5}{\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 = 0.46, size = 112, normalized size = 1.09 \begin {gather*} \frac {15 \, {\left (d x + c\right )} a^{2} + \frac {2 \, {\left (15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 144 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 90 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 65 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.42, size = 105, normalized size = 1.02 \begin {gather*} \frac {3\,a^2\,x}{4}+\frac {\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}+7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {72\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {13\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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