∫ cot−1(c − (1 + ic) cot(a + bx)) dx [181]

Optimal. Leaf size=86 \[ \frac {b x^2}{2}+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {\text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b} \]

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

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Rubi [A]
time = 0.09, antiderivative size = 86, 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 = {5274, 2215, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {b x^2}{2} \end {gather*}

Int[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]

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

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c

+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))

, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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]

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]

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

Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcCot[c + d*Cot[a + b*x]], x] + Dist[I

*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]

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

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Mathematica [A]
time = 0.50, size = 75, normalized size = 0.87 \begin {gather*} x \cot ^{-1}(c+(-1-i c) \cot (a+b x))+\frac {1}{2} i x \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )-\frac {\text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )}{4 b} \end {gather*}

Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1755 vs. \(2 (75 ) = 150\).
time = 0.94, size = 1756, normalized size = 20.42


method	result	size

risch	\(\text {Expression too large to display}\)	\(1244\)
default	\(\text {Expression too large to display}\)	\(1756\)
derivativedivides	\(\text {Expression too large to display}\)	\(1898\)

int(Pi-arccot(-c+(1+I*c)*cot(b*x+a)),x,method=_RETURNVERBOSE)

Pi*x-1/4*I/(1+I*c)/b/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))*c^2+1/4*I/(1+I*c)/b/

(I-c)*ln(-1/2*I*(I-c+(1+I*c)*cot(b*x+a)))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))*c^2-1/4*I/(1+I*c)/b/(I-c)*ln(-c-

(1+I*c)*cot(b*x+a)+I)*ln((-I+c-(1+I*c)*cot(b*x+a))/(-2*I+2*c))*c^2+1/4*I/(1+I*c)/b/(I-c)*ln(-c-(1+I*c)*cot(b*x

+a)+I)*ln(1/2*(I+c-(1+I*c)*cot(b*x+a))/c)*c^2-2*I/(1+I*c)/b*arccot(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(1+

I*c)*cot(b*x+a))*c+2*I/(1+I*c)/b*arccot(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*c-1/2/(1+

I*c)/b/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))*c+1/2/(1+I*c)/b/(I-c)*ln(-1/2*I*(I

-c+(1+I*c)*cot(b*x+a)))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))*c-1/2/(1+I*c)/b/(I-c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*

ln((-I+c-(1+I*c)*cot(b*x+a))/(-2*I+2*c))*c+1/2/(1+I*c)/b/(I-c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*ln(1/2*(I+c-(1+I*c)

*cot(b*x+a))/c)*c+1/(1+I*c)/b*arccot(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(1+I*c)*cot(b*x+a))*c^2-1/(1+I*c)

/b*arccot(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*c^2+1/4*I/(1+I*c)/b/(I-c)*dilog(1/2*(I+

c-(1+I*c)*cot(b*x+a))/c)*c^2+1/4*I/(1+I*c)/b/(I-c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*ln((-I+c-(1+I*c)*cot(b*x+a))/(-

2*I+2*c))-1/4*I/(1+I*c)/b/(I-c)*ln(-c-(1+I*c)*cot(b*x+a)+I)*ln(1/2*(I+c-(1+I*c)*cot(b*x+a))/c)+1/4*I/(1+I*c)/b

/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))-1/4*I/(1+I*c)/b/(I-c)*ln(-1/2*I*(I-c+(1+

I*c)*cot(b*x+a)))*ln(-1/2*I*(I+c-(1+I*c)*cot(b*x+a)))+1/8*I/(1+I*c)/b/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))^2*c^2+1

/4*I/(1+I*c)/b/(I-c)*dilog(-1/2*I*(I-c+(1+I*c)*cot(b*x+a)))*c^2-1/4*I/(1+I*c)/b/(I-c)*dilog((-I+c-(1+I*c)*cot(

b*x+a))/(-2*I+2*c))*c^2+1/2/(1+I*c)/b/(I-c)*dilog(-1/2*I*(I-c+(1+I*c)*cot(b*x+a)))*c-1/2/(1+I*c)/b/(I-c)*dilog

((-I+c-(1+I*c)*cot(b*x+a))/(-2*I+2*c))*c+1/2/(1+I*c)/b/(I-c)*dilog(1/2*(I+c-(1+I*c)*cot(b*x+a))/c)*c-1/(1+I*c)

/b*arccot(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(1+I*c)*cot(b*x+a))+1/(1+I*c)/b*arccot(-c+(1+I*c)*cot(b*x+a)

)/(2*I-2*c)*ln(-c-(1+I*c)*cot(b*x+a)+I)-1/4*I/(1+I*c)/b/(I-c)*dilog(1/2*(I+c-(1+I*c)*cot(b*x+a))/c)-1/8*I/(1+I

*c)/b/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))^2-1/4*I/(1+I*c)/b/(I-c)*dilog(-1/2*I*(I-c+(1+I*c)*cot(b*x+a)))+1/4*I/(1

+I*c)/b/(I-c)*dilog((-I+c-(1+I*c)*cot(b*x+a))/(-2*I+2*c))+1/4/(1+I*c)/b/(I-c)*ln(I-c+(1+I*c)*cot(b*x+a))^2*c

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="maxima")

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for mor

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Fricas [A]
time = 2.83, size = 116, normalized size = 1.35 \begin {gather*} \frac {2 \, b^{2} x^{2} + 4 \, \pi b x + 2 i \, b x \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 \, a^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \end {gather*}

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="fricas")

1/4*(2*b^2*x^2 + 4*pi*b*x + 2*I*b*x*log((c - I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) - I)) - 2*a^2 - 2*(

-I*b*x - I*a)*log(I*c*e^(2*I*b*x + 2*I*a) + 1) - 2*I*a*log((c*e^(2*I*b*x + 2*I*a) - I)/c) + dilog(-I*c*e^(2*I*

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

integrate(pi-acot(-c+(1+I*c)*cot(b*x+a)),x)

Exception raised: CoercionFailed >> Cannot convert -_t0**2*I + 2*c*exp(2*I*a) - I*exp(2*I*a) of type <class 's

ympy.core.add.Add'> to QQ_I[b,c,_t0,exp(I*a)]

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="giac")

integrate(pi - arccot((I*c + 1)*cot(b*x + a) - c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right ) \,d x \end {gather*}

int(Pi + acot(c - cot(a + b*x)*(c*1i + 1)),x)

int(Pi + acot(c - cot(a + b*x)*(c*1i + 1)), x)

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3.2.81 \(\int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx\) [181]