3.35 \(\int \frac{(c+d x)^m}{(a+i a \cot (e+f x))^2} \, dx\)

Optimal. Leaf size=171 \[ \frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a^2 f}-\frac{i 4^{-m-2} e^{4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )}{a^2 f}+\frac{(c+d x)^{m+1}}{4 a^2 d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.184724, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3729, 2181} \[ \frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a^2 f}-\frac{i 4^{-m-2} e^{4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )}{a^2 f}+\frac{(c+d x)^{m+1}}{4 a^2 d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m/(a + I*a*Cot[e + f*x])^2,x]

[Out]

(c + d*x)^(1 + m)/(4*a^2*d*(1 + m)) + (I*2^(-2 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*f
*(c + d*x))/d])/(a^2*f*(((-I)*f*(c + d*x))/d)^m) - (I*4^(-2 - m)*E^((4*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 +
m, ((-4*I)*f*(c + d*x))/d])/(a^2*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))^2} \, dx &=\int \left (\frac{(c+d x)^m}{4 a^2}-\frac{e^{2 i e+2 i f x} (c+d x)^m}{2 a^2}+\frac{e^{4 i e+4 i f x} (c+d x)^m}{4 a^2}\right ) \, dx\\ &=\frac{(c+d x)^{1+m}}{4 a^2 d (1+m)}+\frac{\int e^{4 i e+4 i f x} (c+d x)^m \, dx}{4 a^2}-\frac{\int e^{2 i e+2 i f x} (c+d x)^m \, dx}{2 a^2}\\ &=\frac{(c+d x)^{1+m}}{4 a^2 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^2 f}-\frac{i 4^{-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^2 f}\\ \end{align*}

Mathematica [A]  time = 2.55082, size = 152, normalized size = 0.89 \[ \frac{(c+d x)^m \left (i 2^{2-m} e^{2 i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )-i 4^{-m} e^{4 i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )+\frac{4 f (c+d x)}{d (m+1)}\right )}{16 a^2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m/(a + I*a*Cot[e + f*x])^2,x]

[Out]

((c + d*x)^m*((4*f*(c + d*x))/(d*(1 + m)) + (I*2^(2 - m)*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d
*x))/d])/(((-I)*f*(c + d*x))/d)^m - (I*E^((4*I)*(e - (c*f)/d))*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(4^m*(((-
I)*f*(c + d*x))/d)^m)))/(16*a^2*f)

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+ia\cot \left ( fx+e \right ) \right ) ^{2}}}\, 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))^2,x)

[Out]

int((d*x+c)^m/(a+I*a*cot(f*x+e))^2,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 (4 \, f x + 4 \, e\right )\,{d x} - 2 \,{\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 (4 \, f x + 4 \, e\right )\,{d x} +{\left (-2 i \, d m - 2 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 )}}{4 \,{\left (a^{2} d m + a^{2} 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72403, size = 360, normalized size = 2.11 \begin{align*} \frac{{\left (4 i \, d m + 4 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 (-i \, d m - 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 ) + 4 \,{\left (d f x + c f\right )}{\left (d x + c\right )}^{m}}{16 \,{\left (a^{2} d f m + a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/16*((4*I*d*m + 4*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)
+ (-I*d*m - 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) + 4*(d*
f*x + c*f)*(d*x + c)^m)/(a^2*d*f*m + a^2*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))**2,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 )}^{2}}\,{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))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^m/(I*a*cot(f*x + e) + a)^2, x)