Optimal. Leaf size=173 \[ -\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{2 d}-\frac {11 a^3 \sin ^3(c+d x) \cos (c+d x)}{8 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {15 a^3 x}{16} \]
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Rubi [A] time = 0.24, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2872, 3770, 3767, 8, 2638, 2635, 2633} \[ -\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{2 d}-\frac {11 a^3 \sin ^3(c+d x) \cos (c+d x)}{8 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {15 a^3 x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2872
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (3 a^9 \csc (c+d x)+a^9 \csc ^2(c+d x)-8 a^9 \sin (c+d x)-6 a^9 \sin ^2(c+d x)+6 a^9 \sin ^3(c+d x)+8 a^9 \sin ^4(c+d x)-3 a^9 \sin ^6(c+d x)-a^9 \sin ^7(c+d x)\right ) \, dx}{a^6}\\ &=a^3 \int \csc ^2(c+d x) \, dx-a^3 \int \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (3 a^3\right ) \int \sin ^6(c+d x) \, dx-\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (6 a^3\right ) \int \sin ^3(c+d x) \, dx-\left (8 a^3\right ) \int \sin (c+d x) \, dx+\left (8 a^3\right ) \int \sin ^4(c+d x) \, dx\\ &=-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac {2 a^3 \cos (c+d x) \sin ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{2} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx-\left (3 a^3\right ) \int 1 \, dx+\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {a^3 \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (6 a^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-3 a^3 x-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{8} \left (15 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx\\ &=-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{16} \left (15 a^3\right ) \int 1 \, dx\\ &=-\frac {15 a^3 x}{16}-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.88, size = 168, normalized size = 0.97 \[ \frac {(a \sin (c+d x)+a)^3 \left (-2100 (c+d x)+455 \sin (2 (c+d x))+245 \sin (4 (c+d x))+35 \sin (6 (c+d x))+9065 \cos (c+d x)+875 \cos (3 (c+d x))+49 \cos (5 (c+d x))-5 \cos (7 (c+d x))+1120 \tan \left (\frac {1}{2} (c+d x)\right )-1120 \cot \left (\frac {1}{2} (c+d x)\right )+6720 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6720 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2240 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 173, normalized size = 1.00 \[ -\frac {280 \, a^{3} \cos \left (d x + c\right )^{7} - 70 \, a^{3} \cos \left (d x + c\right )^{5} - 175 \, a^{3} \cos \left (d x + c\right )^{3} + 840 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 840 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 525 \, a^{3} \cos \left (d x + c\right ) + {\left (80 \, a^{3} \cos \left (d x + c\right )^{7} - 336 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, a^{3} \cos \left (d x + c\right )^{3} + 525 \, a^{3} d x - 1680 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{560 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 290, normalized size = 1.68 \[ -\frac {525 \, {\left (d x + c\right )} a^{3} - 1680 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {280 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 4480 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 980 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 20160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 38080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 49280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32256 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 980 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12992 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2496 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 190, normalized size = 1.10 \[ -\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d}-\frac {5 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8 d}-\frac {15 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 d}-\frac {15 a^{3} x}{16}-\frac {15 a^{3} c}{16 d}+\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {3 a^{3} \cos \left (d x +c \right )}{d}+\frac {3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 186, normalized size = 1.08 \[ -\frac {320 \, a^{3} \cos \left (d x + c\right )^{7} - 224 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 280 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{2240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.38, size = 429, normalized size = 2.48 \[ \frac {3\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {15\,a^3\,\mathrm {atan}\left (\frac {225\,a^6}{64\,\left (\frac {45\,a^6}{4}+\frac {225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}-\frac {45\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {45\,a^6}{4}+\frac {225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{4}-32\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-144\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\frac {111\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{4}-272\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-352\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {113\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}-\frac {1152\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {464\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {624\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+a^3}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\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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