Optimal. Leaf size=98 \[ \frac {(c+d x)^{m+1}}{2 a d (m+1)}+\frac {i 2^{-m-2} e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i f (c+d x)}{d}\right )}{a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3727, 2181} \[ \frac {(c+d x)^{m+1}}{2 a d (m+1)}+\frac {i 2^{-m-2} e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 i f (c+d x)}{d}\right )}{a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3727
Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{a+i a \cot (e+f x)} \, dx &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {\int e^{2 i \left (e+\frac {\pi }{2}+f x\right )} (c+d x)^m \, dx}{2 a}\\ &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {i 2^{-2-m} e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i f (c+d x)}{d}\right )}{a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.29, size = 190, normalized size = 1.94 \[ \frac {2^{-m-2} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^m \left (\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\cos \left (e-\frac {c f}{d}\right )+i \sin \left (e-\frac {c f}{d}\right )\right ) \left (f 2^{m+1} (c+d x) \left (-\frac {i f (c+d x)}{d}\right )^m \left (\cos \left (e-\frac {c f}{d}\right )-i \sin \left (e-\frac {c f}{d}\right )\right )+i d (m+1) \left (\cos \left (e-\frac {c f}{d}\right )+i \sin \left (e-\frac {c f}{d}\right )\right ) \Gamma \left (m+1,-\frac {2 i f (c+d x)}{d}\right )\right )}{a d f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 84, normalized size = 0.86 \[ \frac {{\left (i \, d m + i \, d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-2 i \, d f x - 2 i \, c f}{d}\right ) + 2 \, {\left (d f x + c f\right )} {\left (d x + c\right )}^{m}}{4 \, {\left (a d f m + a d f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{i \, a \cot \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{a +i a \cot \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (d m + d\right )} \int {\left (d x + c\right )}^{m} \cos \left (2 \, f x + 2 \, e\right )\,{d x} + {\left (i \, d m + i \, d\right )} \int {\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right )\,{d x} - e^{\left (m \log \left (d x + c\right ) + \log \left (d x + c\right )\right )}}{2 \, {\left (a d m + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^m}{a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {\left (c + d x\right )^{m}}{\cot {\left (e + f x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________