Optimal. Leaf size=140 \[ \frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {a^2 \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac {9 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {15 a^2 x}{4} \]
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Rubi [A] time = 0.21, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2872, 3770, 3767, 8, 3768, 2635, 2633} \[ \frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {a^2 \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac {9 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {15 a^2 x}{4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
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))^2 \, dx &=\frac {\int \left (-6 a^8-2 a^8 \csc (c+d x)+2 a^8 \csc ^2(c+d x)+a^8 \csc ^3(c+d x)+6 a^8 \sin ^2(c+d x)+2 a^8 \sin ^3(c+d x)-2 a^8 \sin ^4(c+d x)-a^8 \sin ^5(c+d x)\right ) \, dx}{a^6}\\ &=-6 a^2 x+a^2 \int \csc ^3(c+d x) \, dx-a^2 \int \sin ^5(c+d x) \, dx-\left (2 a^2\right ) \int \csc (c+d x) \, dx+\left (2 a^2\right ) \int \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \sin ^3(c+d x) \, dx-\left (2 a^2\right ) \int \sin ^4(c+d x) \, dx+\left (6 a^2\right ) \int \sin ^2(c+d x) \, dx\\ &=-6 a^2 x+\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{2 d}+\frac {1}{2} a^2 \int \csc (c+d x) \, dx-\frac {1}{2} \left (3 a^2\right ) \int \sin ^2(c+d x) \, dx+\left (3 a^2\right ) \int 1 \, dx+\frac {a^2 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (2 a^2\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-3 a^2 x+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{2 d}-\frac {1}{4} \left (3 a^2\right ) \int 1 \, dx\\ &=-\frac {15 a^2 x}{4}+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 5.53, size = 174, normalized size = 1.24 \[ \frac {(a \sin (c+d x)+a)^2 \left (-300 (c+d x)-80 \sin (2 (c+d x))-5 \sin (4 (c+d x))-70 \cos (c+d x)+5 \cos (3 (c+d x))+\cos (5 (c+d x))+80 \tan \left (\frac {1}{2} (c+d x)\right )-80 \cot \left (\frac {1}{2} (c+d x)\right )-10 \csc ^2\left (\frac {1}{2} (c+d x)\right )+10 \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{80 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 199, normalized size = 1.42 \[ \frac {4 \, a^{2} \cos \left (d x + c\right )^{7} - 4 \, a^{2} \cos \left (d x + c\right )^{5} - 75 \, a^{2} d x \cos \left (d x + c\right )^{2} - 20 \, a^{2} \cos \left (d x + c\right )^{3} + 75 \, a^{2} d x + 30 \, a^{2} \cos \left (d x + c\right ) + 15 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 5 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{5} + 5 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 244, normalized size = 1.74 \[ \frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 150 \, {\left (d x + c\right )} a^{2} - 60 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 40 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {5 \, {\left (18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {4 \, {\left (45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 50 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 80 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 80 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 50 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 16 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 199, normalized size = 1.42 \[ -\frac {3 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{10 d}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{2} \cos \left (d x +c \right )}{2 d}-\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {2 a^{2} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {5 a^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d}-\frac {15 a^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{4 d}-\frac {15 a^{2} x}{4}-\frac {15 a^{2} c}{4 d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 191, normalized size = 1.36 \[ \frac {2 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} - 5 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} - 15 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 377, normalized size = 2.69 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {15\,a^2\,\mathrm {atan}\left (\frac {225\,a^4}{4\,\left (\frac {45\,a^4}{2}-\frac {225\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )}+\frac {45\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {45\,a^4}{2}-\frac {225\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )}\right )}{2\,d}-\frac {-14\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+\frac {69\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+40\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+37\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+60\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+37\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+38\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {89\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{10}+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^2}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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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