Optimal. Leaf size=134 \[ -\frac{e^{-i (a+b x)} \text{Hypergeometric2F1}\left (1,-\frac{b}{2 d},1-\frac{b}{2 d},-e^{2 i (c+d x)}\right )}{b}+\frac{e^{i (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{b}{2 d},\frac{b}{2 d}+1,-e^{2 i (c+d x)}\right )}{b}+\frac{e^{-i (a+b x)}}{2 b}-\frac{e^{i (a+b x)}}{2 b} \]
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Rubi [A] time = 0.119283, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4560, 2194, 2251} \[ -\frac{e^{-i (a+b x)} \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right )}{b}+\frac{e^{i (a+b x)} \, _2F_1\left (1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 i (c+d x)}\right )}{b}+\frac{e^{-i (a+b x)}}{2 b}-\frac{e^{i (a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 4560
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int \cos (a+b x) \tan (c+d x) \, dx &=\int \left (-\frac{1}{2} i e^{-i (a+b x)}-\frac{1}{2} i e^{i (a+b x)}+\frac{i e^{-i (a+b x)}}{1+e^{2 i (c+d x)}}+\frac{i e^{i (a+b x)}}{1+e^{2 i (c+d x)}}\right ) \, dx\\ &=-\left (\frac{1}{2} i \int e^{-i (a+b x)} \, dx\right )-\frac{1}{2} i \int e^{i (a+b x)} \, dx+i \int \frac{e^{-i (a+b x)}}{1+e^{2 i (c+d x)}} \, dx+i \int \frac{e^{i (a+b x)}}{1+e^{2 i (c+d x)}} \, dx\\ &=\frac{e^{-i (a+b x)}}{2 b}-\frac{e^{i (a+b x)}}{2 b}-\frac{e^{-i (a+b x)} \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right )}{b}+\frac{e^{i (a+b x)} \, _2F_1\left (1,\frac{b}{2 d};1+\frac{b}{2 d};-e^{2 i (c+d x)}\right )}{b}\\ \end{align*}
Mathematica [A] time = 1.8844, size = 142, normalized size = 1.06 \[ \frac{e^{-i (a+b x)} \left ((b-2 d) \left (2 e^{2 i (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{b}{2 d},\frac{b}{2 d}+1,-e^{2 i (c+d x)}\right )-e^{2 i (a+b x)}-1\right )+2 b e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (1,1-\frac{b}{2 d},2-\frac{b}{2 d},-e^{2 i (c+d x)}\right )\right )}{2 b (b-2 d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.175, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( bx+a \right ) \tan \left ( dx+c \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right ) \tan \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (b x + a\right ) \tan \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (a + b x \right )} \tan{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right ) \tan \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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