Optimal. Leaf size=134 \[ \frac {e^{-i (a+b x)}}{2 b}-\frac {e^{i (a+b x)}}{2 b}-\frac {e^{-i (a+b x)} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};-e^{2 i (c+d x)}\right )}{b}+\frac {e^{i (a+b x)} \, _2F_1\left (1,\frac {b}{2 d};1+\frac {b}{2 d};-e^{2 i (c+d x)}\right )}{b} \]
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Rubi [A]
time = 0.08, 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 = {4656, 2225,
2283} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 4656
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 = 1.71, size = 197, normalized size = 1.47 \begin {gather*} \frac {e^{-i (a-2 c+b x)} \left (b (b+2 d) e^{2 i d x} \left (1+e^{2 i c}\right ) \, _2F_1\left (1,1-\frac {b}{2 d};2-\frac {b}{2 d};-e^{2 i (c+d x)}\right )+(b-2 d) \left ((b+2 d) \left (-1+e^{2 i (a+b x)}\right )-b e^{2 i (a+(b+d) x)} \left (1+e^{2 i c}\right ) \, _2F_1\left (1,1+\frac {b}{2 d};2+\frac {b}{2 d};-e^{2 i (c+d x)}\right )\right )\right )}{\left (b^3-4 b d^2\right ) \left (1+e^{2 i c}\right )}+\frac {\sin (a+b x) \tan (c)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \cos {\left (a + b x \right )} \tan {\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \cos \left (a+b\,x\right )\,\mathrm {tan}\left (c+d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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