Optimal. Leaf size=72 \[ \frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2757, 3770, 3767, 8, 3768} \[ \frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2757
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \sec ^4(c+d x) \, dx &=\int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx\\ &=a^3 \int \sec (c+d x) \, dx+a^3 \int \sec ^4(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (3 a^3\right ) \int \sec (c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {3 a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 5.40, size = 154, normalized size = 2.14 \[ -\frac {a^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^3 \left (-4 \tan (c) \cos (c+d x)-\sec (c) (-20 \sin (2 c+d x)+9 \sin (c+2 d x)+9 \sin (3 c+2 d x)+22 \sin (2 c+3 d x)+50 \sin (d x))+60 \cos ^3(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 98, normalized size = 1.36 \[ \frac {15 \, a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (22 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 106, normalized size = 1.47 \[ \frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 80, normalized size = 1.11 \[ \frac {5 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {11 a^{3} \tan \left (d x +c \right )}{3 d}+\frac {3 a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 111, normalized size = 1.54 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - 9 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 112, normalized size = 1.56 \[ \frac {5\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {40\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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