Optimal. Leaf size=181 \[ \frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {85 a^3 x}{16} \]
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Rubi [A] time = 0.25, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3767, 8, 3768, 3770, 2638, 2635, 2633} \[ \frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {85 a^3 x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (-8 a^9+3 a^9 \csc ^2(c+d x)+a^9 \csc ^3(c+d x)-6 a^9 \sin (c+d x)+6 a^9 \sin ^2(c+d x)+8 a^9 \sin ^3(c+d x)-3 a^9 \sin ^5(c+d x)-a^9 \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=-8 a^3 x+a^3 \int \csc ^3(c+d x) \, dx-a^3 \int \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^5(c+d x) \, dx-\left (6 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (8 a^3\right ) \int \sin ^3(c+d x) \, dx\\ &=-8 a^3 x+\frac {6 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{2} a^3 \int \csc (c+d x) \, dx-\frac {1}{6} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (8 a^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-5 a^3 x-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{8} \left (5 a^3\right ) \int \sin ^2(c+d x) \, dx\\ &=-5 a^3 x-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx\\ &=-\frac {85 a^3 x}{16}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 6.37, size = 664, normalized size = 3.67 \[ -\frac {81 \sin (2 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {3 \sin (4 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\sin (6 (c+d x)) (a \sin (c+d x)+a)^3}{192 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {85 (c+d x) (a \sin (c+d x)+a)^3}{16 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {15 \cos (c+d x) (a \sin (c+d x)+a)^3}{8 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {17 \cos (3 (c+d x)) (a \sin (c+d x)+a)^3}{48 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {3 \cos (5 (c+d x)) (a \sin (c+d x)+a)^3}{80 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {(a \sin (c+d x)+a)^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {(a \sin (c+d x)+a)^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {3 \tan \left (\frac {1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{2 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {3 \cot \left (\frac {1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{2 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{8 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{8 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 = 0.82, size = 212, normalized size = 1.17 \[ \frac {144 \, a^{3} \cos \left (d x + c\right )^{7} + 16 \, a^{3} \cos \left (d x + c\right )^{5} - 1275 \, a^{3} d x \cos \left (d x + c\right )^{2} + 80 \, a^{3} \cos \left (d x + c\right )^{3} + 1275 \, a^{3} d x - 120 \, a^{3} \cos \left (d x + c\right ) - 60 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 60 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 34 \, a^{3} \cos \left (d x + c\right )^{5} - 85 \, a^{3} \cos \left (d x + c\right )^{3} + 255 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 306, normalized size = 1.69 \[ \frac {30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1275 \, {\left (d x + c\right )} a^{3} + 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {30 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (645 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1824 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 645 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 544 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 199, normalized size = 1.10 \[ -\frac {17 a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6 d}-\frac {85 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{24 d}-\frac {85 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 d}-\frac {85 a^{3} x}{16}-\frac {85 a^{3} c}{16 d}+\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{10 d}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d}+\frac {a^{3} \cos \left (d x +c \right )}{2 d}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 239, normalized size = 1.32 \[ \frac {96 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 80 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 360 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 438, normalized size = 2.42 \[ \frac {\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{2}+\frac {95\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{2}+\frac {131\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+109\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-75\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {1043\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{6}-135\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+150\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {887\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6}+\frac {533\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{10}-\frac {115\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {227\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}+\frac {85\,a^3\,\mathrm {atan}\left (\frac {7225\,a^6}{64\,\left (\frac {85\,a^6}{8}+\frac {7225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}-\frac {85\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (\frac {85\,a^6}{8}+\frac {7225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
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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