Optimal. Leaf size=73 \[ \frac {a^4 \sin (c+d x)}{d}+\frac {4 a^4 \tan (c+d x)}{d}+\frac {13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x \]
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Rubi [A] time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3791, 2637, 3770, 3767, 8, 3768} \[ \frac {a^4 \sin (c+d x)}{d}+\frac {4 a^4 \tan (c+d x)}{d}+\frac {13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3767
Rule 3768
Rule 3770
Rule 3791
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (4 a^4+a^4 \cos (c+d x)+6 a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+a^4 \sec ^3(c+d x)\right ) \, dx\\ &=4 a^4 x+a^4 \int \cos (c+d x) \, dx+a^4 \int \sec ^3(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^2(c+d x) \, dx+\left (6 a^4\right ) \int \sec (c+d x) \, dx\\ &=4 a^4 x+\frac {6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 \sin (c+d x)}{d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^4 \int \sec (c+d x) \, dx-\frac {\left (4 a^4\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=4 a^4 x+\frac {13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^4 \sin (c+d x)}{d}+\frac {4 a^4 \tan (c+d x)}{d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 1.38, size = 272, normalized size = 3.73 \[ \frac {1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {4 \sin (c) \cos (d x)}{d}+\frac {4 \cos (c) \sin (d x)}{d}+\frac {16 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {16 \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {1}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {1}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {26 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {26 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+16 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 111, normalized size = 1.52 \[ \frac {16 \, a^{4} d x \cos \left (d x + c\right )^{2} + 13 \, a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 13 \, a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} \cos \left (d x + c\right ) + a^{4}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 129, normalized size = 1.77 \[ \frac {8 \, {\left (d x + c\right )} a^{4} + 13 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 13 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {2 \, {\left (7 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.83, size = 86, normalized size = 1.18 \[ \frac {a^{4} \sin \left (d x +c \right )}{d}+4 a^{4} x +\frac {4 a^{4} c}{d}+\frac {13 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {4 a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 110, normalized size = 1.51 \[ \frac {16 \, {\left (d x + c\right )} a^{4} - a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, a^{4} \sin \left (d x + c\right ) + 16 \, a^{4} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 115, normalized size = 1.58 \[ 4\,a^4\,x+\frac {13\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {5\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-11\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ a^{4} \left (\int 4 \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \cos {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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