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

Optimal. Leaf size=213 \[ \frac {b (-2 a c f-2 a d f x+b d) \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac {b (c+d x)}{f \left (a^2+b^2\right ) (a+b \cot (e+f x))}-\frac {(c+d x)^2}{2 d \left (a^2+b^2\right )}+\frac {i a b d \text {Li}_2\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac {(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-i b)^2 (a+i b)} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [B]  time = 7.10, size = 730, normalized size = 3.43 \[ -\frac {2 a c \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}+\frac {d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 \left (\frac {b \left (i \text {Li}_2\left (e^{2 i \left (e+f x+\tan ^{-1}\left (\frac {b}{a}\right )\right )}\right )+i \left (2 \tan ^{-1}\left (\frac {b}{a}\right )-\pi \right ) (e+f x)-2 \left (\tan ^{-1}\left (\frac {b}{a}\right )+e+f x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {b}{a}\right )+e+f x\right )}\right )+2 \tan ^{-1}\left (\frac {b}{a}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {b}{a}\right )+e+f x\right )\right )-\pi \log \left (1+e^{-2 i (e+f x)}\right )+\pi \log (\cos (e+f x))\right )}{a \sqrt {\frac {b^2}{a^2}+1}}+e^{i \tan ^{-1}\left (\frac {b}{a}\right )} (e+f x)^2\right )}{f^2 (b-i a) (b+i a) \sqrt {\frac {a^2+b^2}{a^2}} (a+b \cot (e+f x))^2}+\frac {b d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}+\frac {2 a d e \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left (a^2+b^2\right ) (a+b \cot (e+f x))^2}-\frac {(e+f x) \csc ^2(e+f x) (2 c f+d (e+f x)-2 d e) (a \sin (e+f x)+b \cos (e+f x))^2}{2 f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2}+\frac {\csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x)) (b c f \sin (e+f x)-b d e \sin (e+f x)+b d (e+f x) \sin (e+f x))}{f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((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)/(((-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)*(-P
i + 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 +
ArcTan[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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fricas [B]  time = 0.65, size = 1053, normalized size = 4.94 \[ \text {result too large to display} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x + c}{{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [B]  time = 2.56, size = 990, normalized size = 4.65 \[ \frac {d \,x^{2}}{4 i a b +2 a^{2}-2 b^{2}}+\frac {c x}{2 i a b +a^{2}-b^{2}}+\frac {2 i b a d \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) x}{\left (-i a +b \right )^{2} f \left (i a +b \right ) \left (-i b +a \right )}-\frac {2 i b \,a^{2} c \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{\left (-i a +b \right )^{2} f \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}-\frac {b^{3} d \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}+\frac {2 i b \,a^{2} d e \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}-\frac {4 i b a d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right )}+\frac {2 b^{2} a c \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{\left (-i a +b \right )^{2} f \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}+\frac {4 i b a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (-i a +b \right )^{2} f \left (i a +b \right ) \left (i b -a \right )}+\frac {i b^{2} d \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right ) a}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}-\frac {2 b^{2} a d e \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right ) \left (i b +a \right )}+\frac {2 i b^{2} \left (d x +c \right )}{\left (i a +b \right ) f \left (-i a +b \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )} b -i {\mathrm e}^{2 i \left (f x +e \right )} a +b +i a \right )}-\frac {2 i b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (i b -a \right )}+\frac {2 i b a d \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) e}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (-i b +a \right )}+\frac {2 b a d \,x^{2}}{\left (-i a +b \right )^{2} \left (i a +b \right ) \left (-i b +a \right )}+\frac {4 b a d e x}{\left (-i a +b \right )^{2} f \left (i a +b \right ) \left (-i b +a \right )}+\frac {2 b a d \,e^{2}}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (-i b +a \right )}+\frac {b a d \polylog \left (2, \frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right )}{\left (-i a +b \right )^{2} f^{2} \left (i a +b \right ) \left (-i b +a \right )} \]

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+2*I/(b-I*a)^2/f/(I*a+b)*b*a*d/(a-I*b)*ln(1-(a+I*b)*exp(2*I
*(f*x+e))/(a-I*b))*x-2*I/(b-I*a)^2/f/(I*a+b)*b*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)-1/(b-I*a)^2/f^2/(I*a+b)*b^3*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-I*a)^2/f^2/(I*a+b)*b*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)-4*I/(b-I*a)^2
/f^2/(I*a+b)*b*a*d*e/(I*b-a)*ln(exp(I*(f*x+e)))+2/(b-I*a)^2/f/(I*a+b)*b^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)^2/f/(I*a+b)*b*a*c/(I*b-a)*ln(exp(I*(f*x+e)))+I/(b-I*a)^2/f^2/(I*
a+b)*b^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/(b-I*a)^2/f^2/(I*a+b)*b^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)+2*I*b^2*(d*x+c)/(I*a+b)/f/(b-I*a)^2/(exp(
2*I*(f*x+e))*b-I*exp(2*I*(f*x+e))*a+b+I*a)-2*I/(b-I*a)^2/f^2/(I*a+b)*b^2*d/(I*b-a)*ln(exp(I*(f*x+e)))+2*I/(b-I
*a)^2/f^2/(I*a+b)*b*a*d/(a-I*b)*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*e+2/(b-I*a)^2/(I*a+b)*b*a*d/(a-I*b)*x^2
+4/(b-I*a)^2/f/(I*a+b)*b*a*d/(a-I*b)*e*x+2/(b-I*a)^2/f^2/(I*a+b)*b*a*d/(a-I*b)*e^2+1/(b-I*a)^2/f^2/(I*a+b)*b*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 = 2.03, size = 1181, normalized size = 5.54 \[ \text {result too large to display} \]

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 + b
^3)*c*f - ((4*I*a^2*b + 4*a*b^2)*c*f - 2*(I*a*b^2 + b^3)*d + ((-4*I*a^2*b + 4*a*b^2)*c*f - 2*(-I*a*b^2 + b^3)*
d)*cos(2*f*x + 2*e) + (4*(a^2*b + I*a*b^2)*c*f - (2*a*b^2 + 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*c
os(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 - 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)*cos(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 + 4*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {c+d\,x}{{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c + d x}{\left (a + b \cot {\left (e + f x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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