Optimal. Leaf size=30 \[ \frac {a \log (1-\cos (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3872, 2836, 12, 36, 31, 29} \[ \frac {a \log (1-\cos (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc (c+d x) \sec (c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {a}{(-a-x) x} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{(-a-x) x} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{-a-x} \, dx,x,-a \cos (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a \log (1-\cos (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 63, normalized size = 2.10 \[ \frac {a \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 31, normalized size = 1.03 \[ -\frac {a \log \left (-\cos \left (d x + c\right )\right ) - a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 58, normalized size = 1.93 \[ \frac {a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 15, normalized size = 0.50 \[ \frac {a \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 26, normalized size = 0.87 \[ \frac {a \log \left (\cos \left (d x + c\right ) - 1\right ) - a \log \left (\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 17, normalized size = 0.57 \[ \frac {2\,a\,\mathrm {atanh}\left (1-2\,\cos \left (c+d\,x\right )\right )}{d} \]
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 \[ a \left (\int \csc {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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