Optimal. Leaf size=83 \[ \frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos (c+d x)}{d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a x}{2} \]
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Rubi [A] time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2838, 2591, 288, 321, 203, 2592, 302, 206} \[ \frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos (c+d x)}{d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a x}{2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 302
Rule 321
Rule 2591
Rule 2592
Rule 2838
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^3(c+d x) \cot (c+d x) \, dx+a \int \cos ^2(c+d x) \cot ^2(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac {3 a x}{2}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 77, normalized size = 0.93 \[ \frac {a \left (15 \cos (c+d x)+\cos (3 (c+d x))-3 \left (\sin (2 (c+d x))+4 \cot (c+d x)-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6 c+6 d x\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 107, normalized size = 1.29 \[ \frac {3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, a \cos \left (d x + c\right ) + {\left (2 \, a \cos \left (d x + c\right )^{3} - 9 \, a d x + 6 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 142, normalized size = 1.71 \[ -\frac {9 \, {\left (d x + c\right )} a - 6 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 119, normalized size = 1.43 \[ \frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \cos \left (d x +c \right )}{d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {a \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {3 a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {3 a x}{2}-\frac {3 c a}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 90, normalized size = 1.08 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a - 3 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.77, size = 244, normalized size = 2.94 \[ \frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {16\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-a}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {3\,a\,\mathrm {atan}\left (\frac {9\,a^2}{6\,a^2+9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {6\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^2+9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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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