Optimal. Leaf size=73 \[ \frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \tan (c+d x)}{d}+\frac {4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {13 a^4 x}{2} \]
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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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3791, 2637, 2635, 8, 3770, 3767} \[ \frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \tan (c+d x)}{d}+\frac {4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {13 a^4 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 3767
Rule 3770
Rule 3791
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (6 a^4+4 a^4 \cos (c+d x)+a^4 \cos ^2(c+d x)+4 a^4 \sec (c+d x)+a^4 \sec ^2(c+d x)\right ) \, dx\\ &=6 a^4 x+a^4 \int \cos ^2(c+d x) \, dx+a^4 \int \sec ^2(c+d x) \, dx+\left (4 a^4\right ) \int \cos (c+d x) \, dx+\left (4 a^4\right ) \int \sec (c+d x) \, dx\\ &=6 a^4 x+\frac {4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^4 \int 1 \, dx-\frac {a^4 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {13 a^4 x}{2}+\frac {4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 1.80, size = 241, normalized size = 3.30 \[ \frac {1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {16 \sin (c) \cos (d x)}{d}+\frac {\sin (2 c) \cos (2 d x)}{d}+\frac {16 \cos (c) \sin (d x)}{d}+\frac {\cos (2 c) \sin (2 d x)}{d}+\frac {4 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {16 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {16 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+26 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 105, normalized size = 1.44 \[ \frac {13 \, a^{4} d x \cos \left (d x + c\right ) + 4 \, a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} \cos \left (d x + c\right ) + 2 \, a^{4}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 129, normalized size = 1.77 \[ \frac {13 \, {\left (d x + c\right )} a^{4} + 8 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (7 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 86, normalized size = 1.18 \[ \frac {a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {13 a^{4} x}{2}+\frac {13 a^{4} c}{2 d}+\frac {4 a^{4} \sin \left (d x +c \right )}{d}+\frac {4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{4} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 85, normalized size = 1.16 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 24 \, {\left (d x + c\right )} a^{4} + 8 \, a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{4} \sin \left (d x + c\right ) + 4 \, a^{4} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 117, normalized size = 1.60 \[ \frac {13\,a^4\,x}{2}+\frac {8\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {-5\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+11\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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 \[ a^{4} \left (\int 4 \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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