Optimal. Leaf size=98 \[ \frac{(c+d x)^{m+1}}{2 a d (m+1)}+\frac{i 2^{-m-2} e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )}{a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.122808, antiderivative size = 98, normalized size of antiderivative = 1., 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 3727
Rule 2181
Rubi steps
\begin{align*} \int \frac{(c+d x)^m}{a+i a \tan (e+f x)} \, dx &=\frac{(c+d x)^{1+m}}{2 a d (1+m)}+\frac{\int e^{-2 i (e+f x)} (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 [B] time = 1.22683, size = 205, normalized size = 2.09 \[ \frac{2^{-m-2} (c+d x)^m \sec (e+f x) \left (-\frac{i f (c+d x)}{d}\right )^m \left (\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\sin \left (f \left (\frac{c}{d}+x\right )\right )-i \cos \left (f \left (\frac{c}{d}+x\right )\right )\right ) \left (d (m+1) \left (\sin \left (e-\frac{c f}{d}\right )+i \cos \left (e-\frac{c f}{d}\right )\right ) \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )+f 2^{m+1} (c+d x) \left (\frac{i f (c+d x)}{d}\right )^m \left (\cos \left (e-\frac{c f}{d}\right )+i \sin \left (e-\frac{c f}{d}\right )\right )\right )}{a d f (m+1) (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.197, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{a+ia\tan \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.63216, size = 207, normalized size = 2.11 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{m}}{i \, a \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]