Optimal. Leaf size=138 \[ -\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {33 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {3 a^3 x}{2} \]
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Rubi [A] time = 0.19, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2635} \[ -\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {33 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {3 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (a^7-5 a^7 \csc (c+d x)-5 a^7 \csc ^2(c+d x)+a^7 \csc ^3(c+d x)+3 a^7 \csc ^4(c+d x)+a^7 \csc ^5(c+d x)+3 a^7 \sin (c+d x)+a^7 \sin ^2(c+d x)\right ) \, dx}{a^4}\\ &=a^3 x+a^3 \int \csc ^3(c+d x) \, dx+a^3 \int \csc ^5(c+d x) \, dx+a^3 \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \sin (c+d x) \, dx-\left (5 a^3\right ) \int \csc (c+d x) \, dx-\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx\\ &=a^3 x+\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{2} a^3 \int \csc (c+d x) \, dx+\frac {1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (5 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=\frac {3 a^3 x}{2}+\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {3 a^3 x}{2}+\frac {33 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 215, normalized size = 1.56 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (96 (c+d x)-16 \sin (2 (c+d x))-192 \cos (c+d x)-96 \tan \left (\frac {1}{2} (c+d x)\right )+96 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )-14 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+14 \sec ^2\left (\frac {1}{2} (c+d x)\right )-264 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+264 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+64 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{64 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 231, normalized size = 1.67 \[ \frac {24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 48 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 66 \, a^{3} \cos \left (d x + c\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 8 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 4 \, a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 241, normalized size = 1.75 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, {\left (d x + c\right )} a^{3} - 264 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {64 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {550 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 215, normalized size = 1.56 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {3 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a^{3} x}{2}+\frac {3 a^{3} c}{2 d}-\frac {11 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {11 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}-\frac {33 a^{3} \cos \left (d x +c \right )}{8 d}-\frac {33 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{d}+\frac {3 a^{3} \cot \left (d x +c \right )}{d}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 209, normalized size = 1.51 \[ -\frac {8 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.78, size = 329, normalized size = 2.38 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {33\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {3\,a^3\,\mathrm {atan}\left (\frac {9\,a^6}{\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {99\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+100\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-26\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )} \]
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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