Optimal. Leaf size=134 \[ \frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {7 a^3 x}{2} \]
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Rubi [A] time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2709, 3770, 3767, 8, 3768, 2638, 2635, 2633} \[ \frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {7 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (-5 a^7-5 a^7 \csc (c+d x)+a^7 \csc ^2(c+d x)+3 a^7 \csc ^3(c+d x)+a^7 \csc ^4(c+d x)+a^7 \sin (c+d x)+3 a^7 \sin ^2(c+d x)+a^7 \sin ^3(c+d x)\right ) \, dx}{a^4}\\ &=-5 a^3 x+a^3 \int \csc ^2(c+d x) \, dx+a^3 \int \csc ^4(c+d x) \, dx+a^3 \int \sin (c+d x) \, dx+a^3 \int \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \sin ^2(c+d x) \, dx-\left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-5 a^3 x+\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{2} \left (3 a^3\right ) \int \csc (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-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {7 a^3 x}{2}+\frac {7 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {2 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 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 {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 6.13, size = 201, normalized size = 1.50 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (-84 (c+d x)-18 \sin (2 (c+d x))-42 \cos (c+d x)+2 \cos (3 (c+d x))+20 \tan \left (\frac {1}{2} (c+d x)\right )-20 \cot \left (\frac {1}{2} (c+d x)\right )-9 \csc ^2\left (\frac {1}{2} (c+d x)\right )+9 \sec ^2\left (\frac {1}{2} (c+d x)\right )-84 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+84 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {1}{2} \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+8 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{24 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 206, normalized size = 1.54 \[ \frac {18 \, a^{3} \cos \left (d x + c\right )^{5} - 56 \, a^{3} \cos \left (d x + c\right )^{3} + 42 \, a^{3} \cos \left (d x + c\right ) + 21 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 21 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 2 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 21 \, a^{3} d x \cos \left (d x + c\right )^{2} - 14 \, a^{3} \cos \left (d x + c\right )^{3} + 21 \, a^{3} d x + 21 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 250, normalized size = 1.87 \[ \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 252 \, {\left (d x + c\right )} a^{3} - 252 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 63 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {154 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 153 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 291 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 192 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 414 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 167 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{72 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 190, normalized size = 1.42 \[ -\frac {7 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d}-\frac {7 a^{3} \cos \left (d x +c \right )}{2 d}-\frac {7 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {3 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {9 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {7 a^{3} x}{2}-\frac {7 a^{3} c}{2 d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{3} \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 185, normalized size = 1.38 \[ \frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 18 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} + 4 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + 9 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.78, size = 339, normalized size = 2.53 \[ \frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {7\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {7\,a^3\,\mathrm {atan}\left (\frac {49\,a^6}{49\,a^6-49\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {49\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{49\,a^6-49\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {64\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+46\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {107\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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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