Optimal. Leaf size=51 \[ \frac {a \cos (c+d x)}{d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a x}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2838, 2592, 321, 206, 2635, 8} \[ \frac {a \cos (c+d x)}{d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 321
Rule 2592
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \, dx+a \int \cos (c+d x) \cot (c+d x) \, dx\\ &=\frac {a \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a \int 1 \, dx-\frac {a \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a x}{2}+\frac {a \cos (c+d x)}{d}+\frac {a \cos (c+d x) \sin (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 {a x}{2}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 74, normalized size = 1.45 \[ \frac {a (c+d x)}{2 d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \cos (c+d x)}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 60, normalized size = 1.18 \[ \frac {a d x + a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, a \cos \left (d x + c\right ) - a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 87, normalized size = 1.71 \[ \frac {{\left (d x + c\right )} a + 2 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 63, normalized size = 1.24 \[ \frac {a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a x}{2}+\frac {c a}{2 d}+\frac {a \cos \left (d x +c \right )}{d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 57, normalized size = 1.12 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a + 2 \, a {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.72, size = 160, normalized size = 3.14 \[ \frac {-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,a}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {a\,\mathrm {atan}\left (\frac {a^2}{2\,a^2-a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2-a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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