Optimal. Leaf size=176 \[ \frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {25 a^3 x}{8} \]
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Rubi [A] time = 0.21, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3770, 3768, 3767, 2638, 2635, 8, 2633} \[ \frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {25 a^3 x}{8} \]
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 ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (-6 a^9-8 a^9 \csc (c+d x)+3 a^9 \csc ^3(c+d x)+a^9 \csc ^4(c+d x)+6 a^9 \sin (c+d x)+8 a^9 \sin ^2(c+d x)-3 a^9 \sin ^4(c+d x)-a^9 \sin ^5(c+d x)\right ) \, dx}{a^6}\\ &=-6 a^3 x+a^3 \int \csc ^4(c+d x) \, dx-a^3 \int \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^4(c+d x) \, dx+\left (6 a^3\right ) \int \sin (c+d x) \, dx-\left (8 a^3\right ) \int \csc (c+d x) \, dx+\left (8 a^3\right ) \int \sin ^2(c+d x) \, dx\\ &=-6 a^3 x+\frac {8 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {6 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {4 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{4} \left (9 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^3\right ) \int 1 \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {a^3 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-2 a^3 x+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx\\ &=-\frac {25 a^3 x}{8}+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.43, size = 219, normalized size = 1.24 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (-1500 (c+d x)-600 \sin (2 (c+d x))-45 \sin (4 (c+d x))-2580 \cos (c+d x)-50 \cos (3 (c+d x))+6 \cos (5 (c+d x))+160 \tan \left (\frac {1}{2} (c+d x)\right )-160 \cot \left (\frac {1}{2} (c+d x)\right )-180 \csc ^2\left (\frac {1}{2} (c+d x)\right )+180 \sec ^2\left (\frac {1}{2} (c+d x)\right )-3120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-10 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+160 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{480 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 231, normalized size = 1.31 \[ \frac {90 \, a^{3} \cos \left (d x + c\right )^{7} + 75 \, a^{3} \cos \left (d x + c\right )^{5} - 500 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} \cos \left (d x + c\right ) + 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (24 \, a^{3} \cos \left (d x + c\right )^{7} - 104 \, a^{3} \cos \left (d x + c\right )^{5} - 375 \, a^{3} d x \cos \left (d x + c\right )^{2} - 520 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} d x + 780 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 292, normalized size = 1.66 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 375 \, {\left (d x + c\right )} a^{3} - 780 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {5 \, {\left (286 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 656 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 223, normalized size = 1.27 \[ -\frac {13 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{10 d}-\frac {13 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d}-\frac {13 a^{3} \cos \left (d x +c \right )}{2 d}-\frac {13 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {5 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}-\frac {5 a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}-\frac {25 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{12 d}-\frac {25 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}-\frac {25 a^{3} x}{8}-\frac {25 a^{3} c}{8 d}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 246, normalized size = 1.40 \[ \frac {4 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 30 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 45 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} + 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.99, size = 429, normalized size = 2.44 \[ \frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {13\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-43\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+99\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {86\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+399\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {95\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {1562\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {232\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1114\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {193\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1537\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {14\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {25\,a^3\,\mathrm {atan}\left (\frac {625\,a^6}{16\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,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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