3.36 \(\int \frac{(c+d x)^m}{(a+i a \cot (e+f x))^3} \, dx\)

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]

(c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) + ((3*I)*2^(-4 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*
I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) - ((3*I)*2^(-5 - 2*m)*E^((4*I)*(e - (c*f)/d))*(c + d*x)^m
*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) + (I*2^(-4 - m)*3^(-1 - m)*E^((6*I)*(e
- (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-6*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m)

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Rubi [A]  time = 0.242452, 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]

Int[(c + d*x)^m/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) + ((3*I)*2^(-4 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*
I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) - ((3*I)*2^(-5 - 2*m)*E^((4*I)*(e - (c*f)/d))*(c + d*x)^m
*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) + (I*2^(-4 - m)*3^(-1 - m)*E^((6*I)*(e
- (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-6*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m)

Rule 3729

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{(c+d x)^m}{(a+i a \cot (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 = 3.24539, size = 238, normalized size = 0.95 \[ \frac{2^{-2 m-5} 3^{-m-1} e^{-\frac{6 i c f}{d}} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \left (i d 2^{m+1} 3^{m+2} (m+1) e^{\frac{4 i c f}{d}+2 i e} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )-i d 3^{m+2} (m+1) e^{2 i \left (\frac{c f}{d}+2 e\right )} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )+i d e^{6 i e} 2^{m+1} (m+1) \text{Gamma}\left (m+1,-\frac{6 i f (c+d x)}{d}\right )+f 12^{m+1} e^{\frac{6 i c f}{d}} (c+d x) \left (-\frac{i f (c+d x)}{d}\right )^m\right )}{a^3 d f (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(2^(-5 - 2*m)*3^(-1 - m)*(c + d*x)^m*(12^(1 + m)*E^(((6*I)*c*f)/d)*f*(c + d*x)*(((-I)*f*(c + d*x))/d)^m + I*2^
(1 + m)*3^(2 + m)*d*E^((2*I)*e + ((4*I)*c*f)/d)*(1 + m)*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d] - I*3^(2 + m)*d*E
^((2*I)*(2*e + (c*f)/d))*(1 + m)*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d] + I*2^(1 + m)*d*E^((6*I)*e)*(1 + m)*Gamm
a[1 + m, ((-6*I)*f*(c + d*x))/d]))/(a^3*d*E^(((6*I)*c*f)/d)*f*(1 + m)*(((-I)*f*(c + d*x))/d)^m)

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Maple [F]  time = 0.466, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+ia\cot \left ( fx+e \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m/(a+I*a*cot(f*x+e))^3,x)

[Out]

int((d*x+c)^m/(a+I*a*cot(f*x+e))^3,x)

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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]

integrate((d*x+c)^m/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/8*((d*m + d)*integrate((d*x + c)^m*cos(6*f*x + 6*e), x) - 3*(d*m + d)*integrate((d*x + c)^m*cos(4*f*x + 4*e
), x) + 3*(d*m + d)*integrate((d*x + c)^m*cos(2*f*x + 2*e), x) + (I*d*m + I*d)*integrate((d*x + c)^m*sin(6*f*x
+ 6*e), x) + (-3*I*d*m - 3*I*d)*integrate((d*x + c)^m*sin(4*f*x + 4*e), x) + (3*I*d*m + 3*I*d)*integrate((d*x
+ c)^m*sin(2*f*x + 2*e), x) - e^(m*log(d*x + c) + log(d*x + c)))/(a^3*d*m + a^3*d)

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Fricas [A]  time = 1.77955, size = 513, normalized size = 2.04 \begin{align*} \frac{{\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 ) +{\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 (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 ) + 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]

integrate((d*x+c)^m/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

1/96*((18*I*d*m + 18*I*d)*e^(-(d*m*log(-2*I*f/d) - 2*I*d*e + 2*I*c*f)/d)*gamma(m + 1, (-2*I*d*f*x - 2*I*c*f)/d
) + (-9*I*d*m - 9*I*d)*e^(-(d*m*log(-4*I*f/d) - 4*I*d*e + 4*I*c*f)/d)*gamma(m + 1, (-4*I*d*f*x - 4*I*c*f)/d) +
(2*I*d*m + 2*I*d)*e^(-(d*m*log(-6*I*f/d) - 6*I*d*e + 6*I*c*f)/d)*gamma(m + 1, (-6*I*d*f*x - 6*I*c*f)/d) + 12*
(d*f*x + c*f)*(d*x + c)^m)/(a^3*d*f*m + a^3*d*f)

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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]

integrate((d*x+c)**m/(a+I*a*cot(f*x+e))**3,x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{m}}{{\left (i \, a \cot \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^m/(I*a*cot(f*x + e) + a)^3, x)