Optimal. Leaf size=138 \[ -\frac {a^3 \sin ^3(c+d x)}{d}-\frac {2 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {33 a^3 x}{8} \]
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Rubi [A] time = 0.23, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3872, 2872, 2637, 2635, 8, 2633, 3770, 3767, 3768} \[ -\frac {a^3 \sin ^3(c+d x)}{d}-\frac {2 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {33 a^3 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sin (c+d x) \tan ^3(c+d x) \, dx\\ &=-\frac {\int \left (5 a^7+5 a^7 \cos (c+d x)-a^7 \cos ^2(c+d x)-3 a^7 \cos ^3(c+d x)-a^7 \cos ^4(c+d x)-a^7 \sec (c+d x)-3 a^7 \sec ^2(c+d x)-a^7 \sec ^3(c+d x)\right ) \, dx}{a^4}\\ &=-5 a^3 x+a^3 \int \cos ^2(c+d x) \, dx+a^3 \int \cos ^4(c+d x) \, dx+a^3 \int \sec (c+d x) \, dx+a^3 \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^3(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx-\left (5 a^3\right ) \int \cos (c+d x) \, dx\\ &=-5 a^3 x+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{2} a^3 \int \sec (c+d x) \, dx+\frac {1}{4} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac {9 a^3 x}{2}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{8} \left (3 a^3\right ) \int 1 \, dx\\ &=-\frac {33 a^3 x}{8}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 114, normalized size = 0.83 \[ \frac {a^3 \sec ^2(c+d x) \left (-16 \sin (c+d x)+225 \sin (2 (c+d x))-72 \sin (3 (c+d x))+18 \sin (4 (c+d x))+8 \sin (5 (c+d x))+\sin (6 (c+d x))-264 (c+d x) \cos (2 (c+d x))+192 \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))-264 c-264 d x\right )}{128 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 152, normalized size = 1.10 \[ -\frac {33 \, a^{3} d x \cos \left (d x + c\right )^{2} - 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} + 8 \, a^{3} \cos \left (d x + c\right )^{4} + 7 \, a^{3} \cos \left (d x + c\right )^{3} - 24 \, a^{3} \cos \left (d x + c\right )^{2} + 24 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 180, normalized size = 1.30 \[ -\frac {33 \, {\left (d x + c\right )} a^{3} - 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {8 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 81 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 79 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 159, normalized size = 1.15 \[ \frac {11 a^{3} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}+\frac {33 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}-\frac {33 a^{3} x}{8}-\frac {33 a^{3} c}{8 d}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{3} \sin \left (d x +c \right )}{2 d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 182, normalized size = 1.32 \[ -\frac {16 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{3} - {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 48 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{3} + 8 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.97, size = 204, normalized size = 1.48 \[ \frac {3\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {33\,a^3\,x}{8}+\frac {-\frac {45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{4}-\frac {83\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {79\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {27\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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