Optimal. Leaf size=251 \[ \frac{3 i 2^{-m-4} e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )}{a^3 f}+\frac{3 i 2^{-2 m-5} e^{-4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 i f (c+d x)}{d}\right )}{a^3 f}+\frac{i 2^{-m-4} 3^{-m-1} e^{-6 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 i f (c+d x)}{d}\right )}{a^3 f}+\frac{(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.241977, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3729, 2181} \[ \frac{3 i 2^{-m-4} e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )}{a^3 f}+\frac{3 i 2^{-2 m-5} e^{-4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 i f (c+d x)}{d}\right )}{a^3 f}+\frac{i 2^{-m-4} 3^{-m-1} e^{-6 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 i f (c+d x)}{d}\right )}{a^3 f}+\frac{(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3729
Rule 2181
Rubi steps
\begin{align*} \int \frac{(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^m}{8 a^3}+\frac{3 e^{-2 i e-2 i f x} (c+d x)^m}{8 a^3}+\frac{3 e^{-4 i e-4 i f x} (c+d x)^m}{8 a^3}+\frac{e^{-6 i e-6 i f x} (c+d x)^m}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac{\int e^{-6 i e-6 i f x} (c+d x)^m \, dx}{8 a^3}+\frac{3 \int e^{-2 i e-2 i f x} (c+d x)^m \, dx}{8 a^3}+\frac{3 \int e^{-4 i e-4 i f x} (c+d x)^m \, dx}{8 a^3}\\ &=\frac{(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac{3 i 2^{-4-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^3 f}+\frac{3 i 2^{-5-2 m} e^{-4 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{4 i f (c+d x)}{d}\right )}{a^3 f}+\frac{i 2^{-4-m} 3^{-1-m} e^{-6 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{6 i f (c+d x)}{d}\right )}{a^3 f}\\ \end{align*}
Mathematica [A] time = 55.0155, size = 269, normalized size = 1.07 \[ \frac{e^{-3 i e} 2^{-2 m-5} 3^{-m-1} (c+d x)^m \sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac{i f (c+d x)}{d}\right )^{-m} \left (i d 2^{m+1} 3^{m+2} (m+1) e^{2 i \left (\frac{c f}{d}+2 e\right )} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )+i d 3^{m+2} (m+1) e^{\frac{4 i c f}{d}+2 i e} \text{Gamma}\left (m+1,\frac{4 i f (c+d x)}{d}\right )+i d 2^{m+1} (m+1) e^{\frac{6 i c f}{d}} \text{Gamma}\left (m+1,\frac{6 i f (c+d x)}{d}\right )+e^{6 i e} f 12^{m+1} (c+d x) \left (\frac{i f (c+d x)}{d}\right )^m\right )}{d f (m+1) (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.136, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{3}}}\, 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 (6 \, f x + 6 \, e\right )\,{d x} + 3 \,{\left (d m + d\right )} \int{\left (d x + c\right )}^{m} \cos \left (4 \, f x + 4 \, e\right )\,{d x} + 3 \,{\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 (6 \, f x + 6 \, e\right )\,{d x} -{\left (3 i \, d m + 3 i \, d\right )} \int{\left (d x + c\right )}^{m} \sin \left (4 \, f x + 4 \, e\right )\,{d x} -{\left (3 i \, d m + 3 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 )}}{8 \,{\left (a^{3} d m + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74322, size = 504, normalized size = 2.01 \begin{align*} \frac{{\left (2 i \, d m + 2 i \, d\right )} e^{\left (-\frac{d m \log \left (\frac{6 i \, f}{d}\right ) + 6 i \, d e - 6 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{6 i \, d f x + 6 i \, c f}{d}\right ) +{\left (9 i \, d m + 9 i \, d\right )} e^{\left (-\frac{d m \log \left (\frac{4 i \, f}{d}\right ) + 4 i \, d e - 4 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{4 i \, d f x + 4 i \, c f}{d}\right ) +{\left (18 i \, d m + 18 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 ) + 12 \,{\left (d f x + c f\right )}{\left (d x + c\right )}^{m}}{96 \,{\left (a^{3} d f m + a^{3} 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}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]