Optimal. Leaf size=143 \[ -\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {19 a^3 x}{16} \]
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Rubi [A] time = 0.20, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2873, 2635, 8, 2592, 302, 206, 2565, 30, 2568} \[ -\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {19 a^3 x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 206
Rule 302
Rule 2565
Rule 2568
Rule 2592
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (3 a^3 \cos ^4(c+d x)+a^3 \cos ^3(c+d x) \cot (c+d x)+3 a^3 \cos ^4(c+d x) \sin (c+d x)+a^3 \cos ^4(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^3(c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin (c+d x) \, dx\\ &=\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} a^3 \int \cos ^2(c+d x) \, dx+\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {9 a^3 x}{8}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} a^3 \int 1 \, dx-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {19 a^3 x}{16}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 102, normalized size = 0.71 \[ \frac {a^3 \left (735 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))+840 \cos (c+d x)-100 \cos (3 (c+d x))-36 \cos (5 (c+d x))+960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1140 c+1140 d x\right )}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 128, normalized size = 0.90 \[ -\frac {144 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} d x - 240 \, a^{3} \cos \left (d x + c\right ) + 120 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 120 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 38 \, a^{3} \cos \left (d x + c\right )^{3} - 57 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 229, normalized size = 1.60 \[ \frac {285 \, {\left (d x + c\right )} a^{3} + 240 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1200 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1760 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1296 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 176 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 149, normalized size = 1.04 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6 d}+\frac {19 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{24 d}+\frac {19 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 d}+\frac {19 a^{3} x}{16}+\frac {19 a^{3} c}{16 d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{3} \cos \left (d x +c \right )}{d}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 135, normalized size = 0.94 \[ -\frac {576 \, a^{3} \cos \left (d x + c\right )^{5} - 160 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.51, size = 355, normalized size = 2.48 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {19\,a^3\,\mathrm {atan}\left (\frac {361\,a^6}{64\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {19\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {-\frac {29\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {29\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22\,a^3}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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