Optimal. Leaf size=134 \[ -\frac {e^{-i (a+b x)} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};-e^{2 i (c+d x)}\right )}{b}+\frac {e^{i (a+b x)} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;-e^{2 i (c+d x)}\right )}{b}+\frac {e^{-i (a+b x)}}{2 b}-\frac {e^{i (a+b x)}}{2 b} \]
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Rubi [A] time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 2194
Rule 2251
Rule 4560
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*}
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Mathematica [A] time = 1.66, size = 114, normalized size = 0.85 \[ \frac {e^{-i (a+b x)} \left (2 e^{2 i (a+b x)} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;-e^{2 i (c+d x)}\right )-e^{2 i (a+b x)}-2 \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};-e^{2 i (c+d x)}\right )+1\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right ) \tan \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right ) \tan \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.59, size = 0, normalized size = 0.00 \[ \int \cos \left (b x +a \right ) \tan \left (d x +c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right ) \tan \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,\mathrm {tan}\left (c+d\,x\right ) \,d x \]
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 \[ \int \cos {\left (a + b x \right )} \tan {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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