Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3205, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 3205
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(c+d x)}\right )}{a d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.58 \[ \frac {\cos (c+d x) \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d \sqrt {a \cos ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 84, normalized size = 2.71 \[ \left [-\frac {\sqrt {a \cos \left (d x + c\right )^{2}} \log \left (-\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1}\right )}{2 \, a d \cos \left (d x + c\right )}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right )^{2}} \sqrt {-a}}{a}\right )}{a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 31, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\sqrt {-a \sin \left (d x + c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 31, normalized size = 1.00 \[ -\frac {\cos \left (d x +c \right ) \arctanh \left (\cos \left (d x +c \right )\right )}{\sqrt {a \left (\cos ^{2}\left (d x +c \right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 51, normalized size = 1.65 \[ -\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (d x + c\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (d x + c\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (d x + c\right ) \right |}}\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {cot}\left (c+d\,x\right )}{\sqrt {a\,{\cos \left (c+d\,x\right )}^2}} \,d x \]
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 \[ \int \frac {\cot {\left (c + d x \right )}}{\sqrt {a \cos ^{2}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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