3.1.70 \(\int x \text {ArcTan}(c+(-1-i c) \cot (a+b x)) \, dx\) [70]

Optimal. Leaf size=124 \[ -\frac {b x^3}{6}+\frac {1}{2} x^2 \text {ArcTan}(c-(1+i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{8 b^2} \]

[Out]

-1/6*b*x^3-1/2*x^2*arctan(-c+(1+I*c)*cot(b*x+a))-1/4*I*x^2*ln(1+I*c*exp(2*I*a+2*I*b*x))-1/4*x*polylog(2,-I*c*e
xp(2*I*a+2*I*b*x))/b-1/8*I*polylog(3,-I*c*exp(2*I*a+2*I*b*x))/b^2

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Rubi [A]
time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5281, 2215, 2221, 2611, 2320, 6724} \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c-(1+i c) \cot (a+b x))-\frac {i \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{8 b^2}-\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {b x^3}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*x^3) + (x^2*ArcTan[c - (1 + I*c)*Cot[a + b*x]])/2 - (I/4)*x^2*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] - (
x*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/8)*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 5281

Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m +
1)*(ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1))), x] - Dist[I*(b/(f*(m + 1))), Int[(e + f*x)^(m + 1)/(c - I*d - c*
E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int x \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{2} (i b) \int \frac {x^2}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{2} (b c) \int \frac {e^{2 i a+2 i b x} x^2}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{2} i \int x \log \left (1-\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {\int \text {Li}_2\left (\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx}{4 b}\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_2(-i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{8 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 110, normalized size = 0.89 \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c+(-1-i c) \cot (a+b x))-\frac {i \left (2 b^2 x^2 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )+2 i b x \text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )+\text {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*ArcTan[c + (-1 - I*c)*Cot[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x
*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, I/(c*E^((2*I)*(a + b*x)))]))/b^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.51, size = 1498, normalized size = 12.08

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*b*x^3+1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2+1/8*x^2*P
i*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/8*x^2*Pi*csgn(I/(
exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^2-1/4/b^2*polylog(2,-I*exp(2*I*(b*x+a
))*c)*a+1/2/b^2*a*dilog(1+I*exp(I*(b*x+a))*(I*c)^(1/2))+1/2/b^2*a*dilog(1-I*exp(I*(b*x+a))*(I*c)^(1/2))+1/8*x^
2*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^3-1/8*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(
b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^2+1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2
))-1/4*I*x^2*ln(1+I*c*exp(2*I*(b*x+a)))+1/4*I*ln(exp(2*I*(b*x+a))*c-I)*x^2+1/4*Pi*x^2-1/8*Pi*x^2*csgn(I*exp(I*
(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))+1/4*Pi*x^2*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2-1/2*I*x^2*ln
(exp(I*(b*x+a)))-1/8*x^2*Pi*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^3-1/8*Pi*x^2*csgn(I*exp(2*I*(b*x+a)))^3-1/4*I/b
^2*ln(1+I*c*exp(2*I*(b*x+a)))*a^2-1/4*I/b^2*a^2*ln(-exp(2*I*(b*x+a))*c+I)-1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))*
csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))-1/8*x^2*Pi*csgn(I*exp(2
*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^3+1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(
exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2+1/8*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))*
csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))+1/2*I/b*a*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2))*x-1/4*I*x^2*ln(
c-I)-1/8*x^2*Pi*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/8*x^2*Pi*csgn((exp(2*I*(b*x+a))*c-I)/(ex
p(2*I*(b*x+a))-1))^2+1/8*x^2*Pi*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^3-1/8*I*polylog(3,-I*exp(2*I
*(b*x+a))*c)/b^2+1/8*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c-I))*csgn(I*(exp(2*I*(b*x+a
))*c-I)/(exp(2*I*(b*x+a))-1))-1/8*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I))*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I
*(b*x+a))-1))^2+1/8*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^2+1/8*x^2*Pi*csgn(I
*(c-I))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^2+1/8*x^2*Pi*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^3-1/
2*I/b*ln(1+I*c*exp(2*I*(b*x+a)))*x*a+1/2*I/b*a*ln(1+I*exp(I*(b*x+a))*(I*c)^(1/2))*x-1/4*x*polylog(2,-I*exp(2*I
*(b*x+a))*c)/b+1/2*I/b^2*a^2*ln(1+I*exp(I*(b*x+a))*(I*c)^(1/2))-1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(
2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))-1/8*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csg
n(I*(c-I))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (87) = 174\).
time = 0.28, size = 219, normalized size = 1.77 \begin {gather*} \frac {\frac {6 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \arctan \left ({\left (-i \, c - 1\right )} \cot \left (b x + a\right ) + c\right )}{b} - \frac {{\left (-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), -c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c + 1\right )}}{b {\left (c - i\right )}}}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*((b*x + a)^2 - 2*(b*x + a)*a)*arctan((-I*c - 1)*cot(b*x + a) + c)/b - (-4*I*(b*x + a)^3 + 12*I*(b*x +
a)^2*a - 6*I*b*x*dilog(-I*c*e^(2*I*b*x + 2*I*a)) - 6*(-I*(b*x + a)^2 + 2*I*(b*x + a)*a)*arctan2(c*cos(2*b*x +
2*a), -c*sin(2*b*x + 2*a) + 1) + 3*((b*x + a)^2 - 2*(b*x + a)*a)*log(c^2*cos(2*b*x + 2*a)^2 + c^2*sin(2*b*x +
2*a)^2 - 2*c*sin(2*b*x + 2*a) + 1) + 3*polylog(3, -I*c*e^(2*I*b*x + 2*I*a)))*(I*c + 1)/(b*(c - I)))/b

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Fricas [A]
time = 3.41, size = 144, normalized size = 1.16 \begin {gather*} -\frac {4 \, b^{3} x^{3} - 6 i \, b^{2} x^{2} \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 ) + 4 \, a^{3} + 6 \, b x {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 6 i \, a^{2} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + 6 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 3 i \, {\rm polylog}\left (3, -i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(4*b^3*x^3 - 6*I*b^2*x^2*log(-(c*e^(2*I*b*x + 2*I*a) - I)*e^(-2*I*b*x - 2*I*a)/(c - I)) + 4*a^3 + 6*b*x*
dilog(-I*c*e^(2*I*b*x + 2*I*a)) + 6*I*a^2*log((c*e^(2*I*b*x + 2*I*a) - I)/c) + 6*(I*b^2*x^2 - I*a^2)*log(I*c*e
^(2*I*b*x + 2*I*a) + 1) + 3*I*polylog(3, -I*c*e^(2*I*b*x + 2*I*a)))/b^2

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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(-x*atan(-c-(-1-I*c)*cot(b*x+a)),x)

[Out]

Exception raised: CoercionFailed >> Cannot convert _t0**2 - exp(2*I*a) of type <class 'sympy.core.add.Add'> to
QQ_I[x,b,_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(-x*arctan(-c-(-1-I*c)*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(-x*arctan(-(-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 x\,\mathrm {atan}\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(x*atan(c - cot(a + b*x)*(c*1i + 1)),x)

[Out]

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

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