Optimal. Leaf size=130 \[ \frac {e^{-i (a+b x)} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 i (c+d x)}\right )}{b}-\frac {e^{i (a+b x)} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 i (c+d x)}\right )}{b}-\frac {e^{-i (a+b x)}}{2 b}+\frac {e^{i (a+b x)}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 130, 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 = {4558, 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.
[In]
[Out]
Rule 2194
Rule 2251
Rule 4558
Rubi steps
\begin {align*} \int \cos (a+b x) \cot (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\\ &=\frac {1}{2} i \int e^{-i (a+b x)} \, dx+\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 = 1.79, size = 108, normalized size = 0.83 \[ \frac {e^{-i (a+b x)} \left (-2 e^{2 i (a+b x)} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 i (c+d x)}\right )+e^{2 i (a+b x)}+2 \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 i (c+d x)}\right )-1\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right ) \cot \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right ) \cot \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.77, size = 0, normalized size = 0.00 \[ \int \cos \left (b x +a \right ) \cot \left (d x +c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right ) \cot \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,\mathrm {cot}\left (c+d\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (a + b x \right )} \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________