Optimal. Leaf size=54 \[ \frac {2}{a d \sqrt {a+b \sec (c+d x)}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3885, 51, 63, 207} \[ \frac {2}{a d \sqrt {a+b \sec (c+d x)}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 3885
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)^{3/2}} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac {2}{a d \sqrt {a+b \sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \sec (c+d x)\right )}{a d}\\ &=\frac {2}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{a d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2}{a d \sqrt {a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 128, normalized size = 2.37 \[ \frac {\sec (c+d x) \left (\sqrt {a \cos (c+d x)} \sqrt {a \cos (c+d x)+b} \left (\log \left (1-\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}\right )-\log \left (\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}+1\right )\right )+2 a \cos (c+d x)\right )}{a^2 d \sqrt {a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 260, normalized size = 4.81 \[ \left [\frac {4 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right ) + b\right )} \sqrt {a} \log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} + 4 \, {\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right )}{2 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{2} b d\right )}}, \frac {{\left (a \cos \left (d x + c\right ) + b\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right ) + 2 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a^{3} d \cos \left (d x + c\right ) + a^{2} b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 45, normalized size = 0.83 \[ \frac {-\frac {2 \arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{a \sqrt {a +b \sec \left (d x +c \right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 70, normalized size = 1.30 \[ \frac {\frac {\log \left (\frac {\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.95, size = 50, normalized size = 0.93 \[ \frac {2}{a\,d\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{a^{3/2}\,d} \]
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 \[ \int \frac {\tan {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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