Optimal. Leaf size=59 \[ \frac {7 a^3 x}{2}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3876, 2717,
2715, 8, 3855} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 3855
Rule 3876
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \, 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 \begin {gather*} \frac {a^3 \left (14 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 (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 \sin (c+d x)+\sin (2 (c+d x))\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 71, normalized size = 1.20
method | result | size |
derivativedivides | \(\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} \left (d x +c \right )+3 a^{3} \sin \left (d x +c \right )+a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(71\) |
default | \(\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} \left (d x +c \right )+3 a^{3} \sin \left (d x +c \right )+a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(71\) |
risch | \(\frac {7 a^{3} x}{2}-\frac {3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{4 d}\) | \(102\) |
norman | \(\frac {\frac {7 a^{3} x}{2}+\frac {7 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {9 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-7 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 74, normalized size = 1.25 \begin {gather*} \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} + 2 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 65, normalized size = 1.10 \begin {gather*} \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} \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 \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cos ^{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 = 0.48, size = 100, normalized size = 1.69 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.73, size = 88, normalized size = 1.49 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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