3.59 \(\int \frac{c+d x}{(a+b \cot (e+f x))^2} \, dx\)

Optimal. Leaf size=213 \[ \frac{i a b d \text{PolyLog}\left (2,\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac{b (-2 a c f-2 a d f x+b d) \log \left (1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac{b (c+d x)}{f \left (a^2+b^2\right ) (a+b \cot (e+f x))}-\frac{(c+d x)^2}{2 d \left (a^2+b^2\right )}+\frac{(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-i b)^2 (a+i b)} \]

[Out]

-(c + d*x)^2/(2*(a^2 + b^2)*d) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - I*b)^2*(a + I*b)*d*f^2) + (b*(c + d*x
))/((a^2 + b^2)*f*(a + b*Cot[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - ((a + I*b)*E^((2*I)*(e + f*x)
))/(a - I*b)])/((a^2 + b^2)^2*f^2) + (I*a*b*d*PolyLog[2, ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/((a^2 + b
^2)^2*f^2)

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Rubi [A]  time = 0.292467, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3733, 3731, 2190, 2279, 2391} \[ \frac{i a b d \text{PolyLog}\left (2,\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac{b (-2 a c f-2 a d f x+b d) \log \left (1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac{b (c+d x)}{f \left (a^2+b^2\right ) (a+b \cot (e+f x))}-\frac{(c+d x)^2}{2 d \left (a^2+b^2\right )}+\frac{(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-i b)^2 (a+i b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c + d*x)^2/(2*(a^2 + b^2)*d) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - I*b)^2*(a + I*b)*d*f^2) + (b*(c + d*x
))/((a^2 + b^2)*f*(a + b*Cot[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - ((a + I*b)*E^((2*I)*(e + f*x)
))/(a - I*b)])/((a^2 + b^2)^2*f^2) + (I*a*b*d*PolyLog[2, ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/((a^2 + b
^2)^2*f^2)

Rule 3733

Int[((c_.) + (d_.)*(x_))/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> -Simp[(c + d*x)^2/(2*d*(a^2 +
b^2)), x] + (Dist[1/(f*(a^2 + b^2)), Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x], x] - Simp[(b*(c
+ d*x))/(f*(a^2 + b^2)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]

Rule 3731

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^
(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])/((a +
I*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Integer
Q[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{c+d x}{(a+b \cot (e+f x))^2} \, dx &=-\frac{(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac{b (c+d x)}{\left (a^2+b^2\right ) f (a+b \cot (e+f x))}+\frac{\int \frac{-b d+2 a c f+2 a d f x}{a+b \cot (e+f x)} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac{(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac{(b d-2 a c f-2 a d f x)^2}{4 a (a-i b)^2 (a+i b) d f^2}+\frac{b (c+d x)}{\left (a^2+b^2\right ) f (a+b \cot (e+f x))}+\frac{(2 i b) \int \frac{e^{2 i (e+f x)} (-b d+2 a c f+2 a d f x)}{(a-i b)^2+\left (-a^2-b^2\right ) e^{2 i (e+f x)}} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac{(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac{(b d-2 a c f-2 a d f x)^2}{4 a (a-i b)^2 (a+i b) d f^2}+\frac{b (c+d x)}{\left (a^2+b^2\right ) f (a+b \cot (e+f x))}+\frac{b (b d-2 a c f-2 a d f x) \log \left (1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac{(2 a b d) \int \log \left (1+\frac{\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{\left (a^2+b^2\right )^2 f}\\ &=-\frac{(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac{(b d-2 a c f-2 a d f x)^2}{4 a (a-i b)^2 (a+i b) d f^2}+\frac{b (c+d x)}{\left (a^2+b^2\right ) f (a+b \cot (e+f x))}+\frac{b (b d-2 a c f-2 a d f x) \log \left (1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac{(i a b d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\left (-a^2-b^2\right ) x}{(a-i b)^2}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{\left (a^2+b^2\right )^2 f^2}\\ &=-\frac{(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac{(b d-2 a c f-2 a d f x)^2}{4 a (a-i b)^2 (a+i b) d f^2}+\frac{b (c+d x)}{\left (a^2+b^2\right ) f (a+b \cot (e+f x))}+\frac{b (b d-2 a c f-2 a d f x) \log \left (1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac{i a b d \text{Li}_2\left (\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}\\ \end{align*}

Mathematica [B]  time = 7.04824, size = 730, normalized size = 3.43 \[ \frac{d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 \left (\frac{b \left (i \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}\left (\frac{b}{a}\right )+e+f x\right )}\right )+i \left (2 \tan ^{-1}\left (\frac{b}{a}\right )-\pi \right ) (e+f x)-2 \left (\tan ^{-1}\left (\frac{b}{a}\right )+e+f x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{b}{a}\right )+e+f x\right )}\right )+2 \tan ^{-1}\left (\frac{b}{a}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac{b}{a}\right )+e+f x\right )\right )-\pi \log \left (1+e^{-2 i (e+f x)}\right )+\pi \log (\cos (e+f x))\right )}{a \sqrt{\frac{b^2}{a^2}+1}}+e^{i \tan ^{-1}\left (\frac{b}{a}\right )} (e+f x)^2\right )}{f^2 (b-i a) (b+i a) \sqrt{\frac{a^2+b^2}{a^2}} (a+b \cot (e+f x))^2}-\frac{2 a c \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}+\frac{b d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}+\frac{2 a d e \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}-\frac{(e+f x) \csc ^2(e+f x) (2 c f+d (e+f x)-2 d e) (a \sin (e+f x)+b \cos (e+f x))^2}{2 f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2}+\frac{\csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x)) (b c f \sin (e+f x)-b d e \sin (e+f x)+b d (e+f x) \sin (e+f x))}{f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((e + f*x)*(-2*d*e + 2*c*f + d*(e + f*x))*Csc[e + f*x]^2*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(2*((-I)*a + b)
*(I*a + b)*f^2*(a + b*Cot[e + f*x])^2) + (b*d*Csc[e + f*x]^2*(-(a*(e + f*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e
+ f*x]])*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*(a^2 + b^2)*f^2*(a + b*Cot[e + f*x])^2)
+ (2*a*d*e*Csc[e + f*x]^2*(-(a*(e + f*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e + f*x]])*(b*Cos[e + f*x] + a*Sin[e
+ f*x])^2)/(((-I)*a + b)*(I*a + b)*(a^2 + b^2)*f^2*(a + b*Cot[e + f*x])^2) - (2*a*c*Csc[e + f*x]^2*(-(a*(e + f
*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e + f*x]])*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*(a
^2 + b^2)*f*(a + b*Cot[e + f*x])^2) + (d*Csc[e + f*x]^2*(E^(I*ArcTan[b/a])*(e + f*x)^2 + (b*(I*(e + f*x)*(-Pi
+ 2*ArcTan[b/a]) - Pi*Log[1 + E^((-2*I)*(e + f*x))] - 2*(e + f*x + ArcTan[b/a])*Log[1 - E^((2*I)*(e + f*x + Ar
cTan[b/a]))] + Pi*Log[Cos[e + f*x]] + 2*ArcTan[b/a]*Log[Sin[e + f*x + ArcTan[b/a]]] + I*PolyLog[2, E^((2*I)*(e
+ f*x + ArcTan[b/a]))]))/(a*Sqrt[1 + b^2/a^2]))*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*
Sqrt[(a^2 + b^2)/a^2]*f^2*(a + b*Cot[e + f*x])^2) + (Csc[e + f*x]^2*(b*Cos[e + f*x] + a*Sin[e + f*x])*(-(b*d*e
*Sin[e + f*x]) + b*c*f*Sin[e + f*x] + b*d*(e + f*x)*Sin[e + f*x]))/(((-I)*a + b)*(I*a + b)*f^2*(a + b*Cot[e +
f*x])^2)

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Maple [B]  time = 0.53, size = 990, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(a+b*cot(f*x+e))^2,x)

[Out]

1/2/(2*I*a*b+a^2-b^2)*d*x^2+1/(2*I*a*b+a^2-b^2)*c*x+I*b^2/(I*a+b)/f^2/(b-I*a)^2*d/(I*b-a)/(a+I*b)*ln(a*exp(2*I
*(f*x+e))+I*exp(2*I*(f*x+e))*b-a+I*b)*a-2*I*b/(I*a+b)/f/(b-I*a)^2*a^2*c/(I*b-a)/(a+I*b)*ln(a*exp(2*I*(f*x+e))+
I*exp(2*I*(f*x+e))*b-a+I*b)-b^3/(I*a+b)/f^2/(b-I*a)^2*d/(I*b-a)/(a+I*b)*ln(a*exp(2*I*(f*x+e))+I*exp(2*I*(f*x+e
))*b-a+I*b)-2*I*b^2/(I*a+b)/f^2/(b-I*a)^2*d/(I*b-a)*ln(exp(I*(f*x+e)))+2*I*b^2*(d*x+c)/(I*a+b)/f/(b-I*a)^2/(b*
exp(2*I*(f*x+e))-I*exp(2*I*(f*x+e))*a+b+I*a)+2*b^2/(I*a+b)/f/(b-I*a)^2*a*c/(I*b-a)/(a+I*b)*ln(a*exp(2*I*(f*x+e
))+I*exp(2*I*(f*x+e))*b-a+I*b)+4*I*b/(I*a+b)/f/(b-I*a)^2*a*c/(I*b-a)*ln(exp(I*(f*x+e)))+2*I*b/(I*a+b)/f/(b-I*a
)^2*a*d/(a-I*b)*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x-2*b^2/(I*a+b)/f^2/(b-I*a)^2*a*d*e/(I*b-a)/(a+I*b)*ln(
a*exp(2*I*(f*x+e))+I*exp(2*I*(f*x+e))*b-a+I*b)-4*I*b/(I*a+b)/f^2/(b-I*a)^2*a*d*e/(I*b-a)*ln(exp(I*(f*x+e)))+2*
I*b/(I*a+b)/f^2/(b-I*a)^2*a*d/(a-I*b)*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*e+2*I*b/(I*a+b)/f^2/(b-I*a)^2*a^2
*d*e/(I*b-a)/(a+I*b)*ln(a*exp(2*I*(f*x+e))+I*exp(2*I*(f*x+e))*b-a+I*b)+2*b/(I*a+b)/(b-I*a)^2*a*d/(a-I*b)*x^2+4
*b/(I*a+b)/f/(b-I*a)^2*a*d/(a-I*b)*e*x+2*b/(I*a+b)/f^2/(b-I*a)^2*a*d/(a-I*b)*e^2+b/(I*a+b)/f^2/(b-I*a)^2*a*d/(
a-I*b)*polylog(2,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))

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Maxima [B]  time = 3.77396, size = 1598, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(a+b*cot(f*x+e))^2,x, algorithm="maxima")

[Out]

-((a^3 + I*a^2*b + a*b^2 + I*b^3)*d*f^2*x^2 + (2*a^3 + 2*I*a^2*b + 2*a*b^2 + 2*I*b^3)*c*f^2*x - (-4*I*a*b^2 -
4*b^3)*c*f - ((4*I*a^2*b + 4*a*b^2)*c*f + (-2*I*a*b^2 - 2*b^3)*d + ((-4*I*a^2*b + 4*a*b^2)*c*f + (2*I*a*b^2 -
2*b^3)*d)*cos(2*f*x + 2*e) + 2*(2*(a^2*b + I*a*b^2)*c*f - (a*b^2 + I*b^3)*d)*sin(2*f*x + 2*e))*arctan2(b*cos(2
*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, a*cos(2*f*x + 2*e) - b*sin(2*f*x + 2*e) - a) - ((-4*I*a^2*b + 4*a*b^2)*d
*f*x*cos(2*f*x + 2*e) + 4*(a^2*b + I*a*b^2)*d*f*x*sin(2*f*x + 2*e) + (4*I*a^2*b + 4*a*b^2)*d*f*x)*arctan2(-(2*
a*b*cos(2*f*x + 2*e) + (a^2 - b^2)*sin(2*f*x + 2*e))/(a^2 + b^2), (2*a*b*sin(2*f*x + 2*e) + a^2 + b^2 - (a^2 -
b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) - ((a^3 + 3*I*a^2*b - 3*a*b^2 - I*b^3)*d*f^2*x^2 + ((2*a^3 + 6*I*a^2*b -
6*a*b^2 - 2*I*b^3)*c*f^2 + (-4*I*a*b^2 + 4*b^3)*d*f)*x)*cos(2*f*x + 2*e) - ((2*I*a^2*b - 2*a*b^2)*d*cos(2*f*x
+ 2*e) - 2*(a^2*b + I*a*b^2)*d*sin(2*f*x + 2*e) + (-2*I*a^2*b - 2*a*b^2)*d)*dilog((I*a - b)*e^(2*I*f*x + 2*I*e
)/(I*a + b)) - (2*(a^2*b - I*a*b^2)*c*f - (a*b^2 - I*b^3)*d - (2*(a^2*b + I*a*b^2)*c*f - (a*b^2 + I*b^3)*d)*co
s(2*f*x + 2*e) + ((-2*I*a^2*b + 2*a*b^2)*c*f + (I*a*b^2 - b^3)*d)*sin(2*f*x + 2*e))*log((a^2 + b^2)*cos(2*f*x
+ 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 + a^2 + b^2 - 2*(a^2 - b^2)*cos(2*f*x + 2*e
)) + (2*(a^2*b + I*a*b^2)*d*f*x*cos(2*f*x + 2*e) - (-2*I*a^2*b + 2*a*b^2)*d*f*x*sin(2*f*x + 2*e) - 2*(a^2*b -
I*a*b^2)*d*f*x)*log(((a^2 + b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2
+ a^2 + b^2 - 2*(a^2 - b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) - ((I*a^3 - 3*a^2*b - 3*I*a*b^2 + b^3)*d*f^2*x^2 +
((2*I*a^3 - 6*a^2*b - 6*I*a*b^2 + 2*b^3)*c*f^2 + 4*(a*b^2 + I*b^3)*d*f)*x)*sin(2*f*x + 2*e))/((2*a^5 + 2*I*a^4
*b + 4*a^3*b^2 + 4*I*a^2*b^3 + 2*a*b^4 + 2*I*b^5)*f^2*cos(2*f*x + 2*e) - (-2*I*a^5 + 2*a^4*b - 4*I*a^3*b^2 + 4
*a^2*b^3 - 2*I*a*b^4 + 2*b^5)*f^2*sin(2*f*x + 2*e) - (2*a^5 - 2*I*a^4*b + 4*a^3*b^2 - 4*I*a^2*b^3 + 2*a*b^4 -
2*I*b^5)*f^2)

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Fricas [B]  time = 2.21571, size = 2394, normalized size = 11.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(a+b*cot(f*x+e))^2,x, algorithm="fricas")

[Out]

1/2*((a^2*b - b^3)*d*f^2*x^2 - 2*a*b^2*c*f - 2*(a*b^2*d*f - (a^2*b - b^3)*c*f^2)*x + ((a^2*b - b^3)*d*f^2*x^2
- 2*a*b^2*c*f - 2*(a*b^2*d*f - (a^2*b - b^3)*c*f^2)*x)*cos(2*f*x + 2*e) + (I*a*b^2*d*cos(2*f*x + 2*e) + I*a^2*
b*d*sin(2*f*x + 2*e) + I*a*b^2*d)*dilog(-(a^2 + b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (-I*a^2 + 2*a*b
+ I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2) + 1) + (-I*a*b^2*d*cos(2*f*x + 2*e) - I*a^2*b*d*sin(2*f*x + 2*e) - I*a
*b^2*d)*dilog(-(a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^2 + 2*a*b - I*b^2)*sin(2*f*x + 2*e))
/(a^2 + b^2) + 1) + (2*a*b^2*d*e - 2*a*b^2*c*f + b^3*d + (2*a*b^2*d*e - 2*a*b^2*c*f + b^3*d)*cos(2*f*x + 2*e)
+ (2*a^2*b*d*e - 2*a^2*b*c*f + a*b^2*d)*sin(2*f*x + 2*e))*log(1/2*a^2 + I*a*b - 1/2*b^2 - 1/2*(a^2 + b^2)*cos(
2*f*x + 2*e) + 1/2*(I*a^2 + I*b^2)*sin(2*f*x + 2*e)) + (2*a*b^2*d*e - 2*a*b^2*c*f + b^3*d + (2*a*b^2*d*e - 2*a
*b^2*c*f + b^3*d)*cos(2*f*x + 2*e) + (2*a^2*b*d*e - 2*a^2*b*c*f + a*b^2*d)*sin(2*f*x + 2*e))*log(-1/2*a^2 + I*
a*b + 1/2*b^2 + 1/2*(a^2 + b^2)*cos(2*f*x + 2*e) + 1/2*(I*a^2 + I*b^2)*sin(2*f*x + 2*e)) - 2*(a*b^2*d*f*x + a*
b^2*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cos(2*f*x + 2*e) + (a^2*b*d*f*x + a^2*b*d*e)*sin(2*f*x + 2*e))*log((a^2 +
b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (-I*a^2 + 2*a*b + I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2)) - 2*(a*
b^2*d*f*x + a*b^2*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cos(2*f*x + 2*e) + (a^2*b*d*f*x + a^2*b*d*e)*sin(2*f*x + 2*e
))*log((a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^2 + 2*a*b - I*b^2)*sin(2*f*x + 2*e))/(a^2 +
b^2)) + ((a^3 - a*b^2)*d*f^2*x^2 + 2*b^3*c*f + 2*(b^3*d*f + (a^3 - a*b^2)*c*f^2)*x)*sin(2*f*x + 2*e))/((a^4*b
+ 2*a^2*b^3 + b^5)*f^2*cos(2*f*x + 2*e) + (a^5 + 2*a^3*b^2 + a*b^4)*f^2*sin(2*f*x + 2*e) + (a^4*b + 2*a^2*b^3
+ b^5)*f^2)

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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)/(a+b*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{d x + c}{{\left (b \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)/(a+b*cot(f*x+e))^2,x, algorithm="giac")

[Out]

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