Optimal. Leaf size=59 \[ \frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^3 x}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2757, 2637, 2635, 8, 3770} \[ \frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 2757
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \sec (c+d x) \, dx &=\int \left (3 a^3+3 a^3 \cos (c+d x)+a^3 \cos ^2(c+d x)+a^3 \sec (c+d x)\right ) \, dx\\ &=3 a^3 x+a^3 \int \cos ^2(c+d x) \, dx+a^3 \int \sec (c+d x) \, dx+\left (3 a^3\right ) \int \cos (c+d x) \, dx\\ &=3 a^3 x+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^3 \int 1 \, dx\\ &=\frac {7 a^3 x}{2}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 1.37 \[ \frac {a^3 \left (12 \sin (c+d x)+\sin (2 (c+d x))-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+14 d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.64, size = 65, normalized size = 1.10 \[ \frac {7 \, a^{3} d x + a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{3} \cos \left (d x + c\right ) + 6 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 100, normalized size = 1.69 \[ \frac {7 \, {\left (d x + c\right )} a^{3} + 2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 72, normalized size = 1.22 \[ \frac {a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 a^{3} x}{2}+\frac {7 a^{3} c}{2 d}+\frac {3 a^{3} \sin \left (d x +c \right )}{d}+\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 67, normalized size = 1.14 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 12 \, {\left (d x + c\right )} a^{3} + 4 \, a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 88, normalized size = 1.49 \[ \frac {7\,a^3\,x}{2}+\frac {2\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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 \[ a^{3} \left (\int 3 \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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