3.2.77 \(\int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\) [177]

Optimal. Leaf size=85 \[ -\frac {b x^2}{2}+x \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {\text {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b} \]

[Out]

-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
*a+2*I*b*x))/b

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Rubi [A]
time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*x^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
, I*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)

Rule 2215

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]

Rule 2221

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]

Rule 2317

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 2438

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 5274

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]

Rubi steps

\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 = 0.44, size = 75, normalized size = 0.88 \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*}

Antiderivative was successfully verified.

[In]

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

[Out]

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
)*(a + b*x)))]/(4*b)

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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. 1497 vs. \(2 (76 ) = 152\).
time = 0.85, size = 1498, normalized size = 17.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(1244\)
default	\(\text {Expression too large to display}\)	\(1498\)
derivativedivides	\(\text {Expression too large to display}\)	\(1673\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Pi*x+2*I/b/(-1+I*c)*arccot(cot(b*x+a)*(-1+I*c)-c)/(2*I+2*c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*c-2*I/b/(-1+I*c)*arcco
t(cot(b*x+a)*(-1+I*c)-c)/(2*I+2*c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)*c-1/4*I/b/(-1+I*c)/(I+c)*ln(-I+cot(b*x+a)*(-1+
I*c)-c)*ln(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I))*c^2-1/4*I/b/(-1+I*c)/(I+c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*ln((-I+cot
(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))*c^2+1/4*I/b/(-1+I*c)/(I+c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*ln(-1/2*(cot(b*x+a)*(-1
+I*c)-c+I)/c)*c^2-1/b/(-1+I*c)*arccot(cot(b*x+a)*(-1+I*c)-c)/(2*I+2*c)*ln(cot(b*x+a)*(-1+I*c)+c+I)+1/4*I/b/(-1
+I*c)/(I+c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)*ln(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I))+1/4*I/b/(-1+I*c)/(I+c)*ln(cot(b*
x+a)*(-1+I*c)+c+I)*ln((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))-1/4*I/b/(-1+I*c)/(I+c)*ln(cot(b*x+a)*(-1+I*c)+c+I
)*ln(-1/2*(cot(b*x+a)*(-1+I*c)-c+I)/c)+1/8*I/b/(-1+I*c)/(I+c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)^2*c^2-1/4*I/b/(-1+I
*c)/(I+c)*dilog(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I))*c^2-1/4*I/b/(-1+I*c)/(I+c)*dilog((-I+cot(b*x+a)*(-1+I*c)-c)/
(-2*I-2*c))*c^2+1/4*I/b/(-1+I*c)/(I+c)*dilog(-1/2*(cot(b*x+a)*(-1+I*c)-c+I)/c)*c^2+1/b/(-1+I*c)*arccot(cot(b*x
+a)*(-1+I*c)-c)/(2*I+2*c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*c^2-1/b/(-1+I*c)*arccot(cot(b*x+a)*(-1+I*c)-c)/(2*I+2*c)
*ln(-I+cot(b*x+a)*(-1+I*c)-c)*c^2+1/2/b/(-1+I*c)/(I+c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)*ln(-1/2*I*(cot(b*x+a)*(-1+
I*c)-c+I))*c+1/2/b/(-1+I*c)/(I+c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*ln((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))*c-1/2/
b/(-1+I*c)/(I+c)*ln(cot(b*x+a)*(-1+I*c)+c+I)*ln(-1/2*(cot(b*x+a)*(-1+I*c)-c+I)/c)*c+1/b/(-1+I*c)*arccot(cot(b*
x+a)*(-1+I*c)-c)/(2*I+2*c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)-1/4/b/(-1+I*c)/(I+c)*ln(-I+cot(b*x+a)*(-1+I*c)-c)^2*c+
1/2/b/(-1+I*c)/(I+c)*dilog(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I))*c+1/2/b/(-1+I*c)/(I+c)*dilog((-I+cot(b*x+a)*(-1+I
*c)-c)/(-2*I-2*c))*c-1/2/b/(-1+I*c)/(I+c)*dilog(-1/2*(cot(b*x+a)*(-1+I*c)-c+I)/c)*c-1/8*I/b/(-1+I*c)/(I+c)*ln(
-I+cot(b*x+a)*(-1+I*c)-c)^2+1/4*I/b/(-1+I*c)/(I+c)*dilog(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I))+1/4*I/b/(-1+I*c)/(I
+c)*dilog((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))-1/4*I/b/(-1+I*c)/(I+c)*dilog(-1/2*(cot(b*x+a)*(-1+I*c)-c+I)/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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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
e details)Is

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Fricas [A]
time = 2.02, size = 116, normalized size = 1.36 \begin {gather*} -\frac {2 \, b^{2} x^{2} - 4 \, \pi b x - 2 i \, b x \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + 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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: CoercionFailed >> Cannot convert -_t0**4 + 3*_t0**2*I*c*exp(2*I*a) - _t0**2*exp(2*I*a) + 2*c
**2*exp(4*I*a) + I*c*exp(4*I*a) of type <class 'sympy.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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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