Optimal. Leaf size=38 \[ \frac {a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3958, 2611, 3770} \[ \frac {a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rule 3958
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx &=-\left ((a A) \int \cot ^2(c+d x) \csc (c+d x) \, dx\right )\\ &=\frac {a A \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} (a A) \int \csc (c+d x) \, dx\\ &=-\frac {a A \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {a A \cot (c+d x) \csc (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 79, normalized size = 2.08 \[ -a A \left (-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 86, normalized size = 2.26 \[ -\frac {2 \, A a \cos \left (d x + c\right ) + {\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 101, normalized size = 2.66 \[ \frac {2 \, A a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac {A a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {{\left (A a + \frac {2 \, A a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 44, normalized size = 1.16 \[ \frac {a A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{2 d}+\frac {a A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 68, normalized size = 1.79 \[ -\frac {A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 56, normalized size = 1.47 \[ \frac {A\,a\,\left (4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+1\right )}{8\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2} \]
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 \[ - A a \left (\int \left (- \csc {\left (c + d x \right )}\right )\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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