Optimal. Leaf size=31 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.04, 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 = {3885, 63, 207} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 3885
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [B] time = 0.20, size = 108, normalized size = 3.48 \[ \frac {\sqrt {a \cos (c+d x)+b} \left (\log \left (1-\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}\right )-\log \left (\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}+1\right )\right )}{d \sqrt {a \cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.61, size = 145, normalized size = 4.68 \[ \left [\frac {\log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} + 4 \, {\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right )}{2 \, \sqrt {a} d}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right )}{a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.61, size = 109, normalized size = 3.52 \[ -\frac {2 \, \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} d \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 26, normalized size = 0.84 \[ -\frac {2 \arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 49, normalized size = 1.58 \[ \frac {\log \left (\frac {\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 27, normalized size = 0.87 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{\sqrt {a}\,d} \]
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 \[ \int \frac {\tan {\left (c + d x \right )}}{\sqrt {a + b \sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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