3.4.44 \(\int \tanh ^{-1}(1-i d-d \cot (a+b x)) \, dx\) [344]

Optimal. Leaf size=94 \[ \frac {1}{2} i b x^2+x \tanh ^{-1}(1-i d-d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac {i \text {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{4 b} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 94, 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 = {6392, 2215, 2221, 2317, 2438} \begin {gather*} \frac {i \text {Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+x \tanh ^{-1}(d (-\cot (a+b x))-i d+1)+\frac {1}{2} i b x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[1 - I*d - d*Cot[a + b*x]],x]

[Out]

(I/2)*b*x^2 + x*ArcTanh[1 - I*d - d*Cot[a + b*x]] - (x*Log[1 - (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/2 + ((I/4)*
PolyLog[2, (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/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 6392

Int[ArcTanh[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcTanh[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 \tanh ^{-1}(1-i d-d \cot (a+b x)) \, dx &=x \tanh ^{-1}(1-i d-d \cot (a+b x))+(i b) \int \frac {x}{1+(-1+i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac {1}{2} i b x^2+x \tanh ^{-1}(1-i d-d \cot (a+b x))+(b (i+d)) \int \frac {e^{2 i a+2 i b x} x}{1+(-1+i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac {1}{2} i b x^2+x \tanh ^{-1}(1-i d-d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac {1}{2} \int \log \left (1+(-1+i d) e^{2 i a+2 i b x}\right ) \, dx\\ &=\frac {1}{2} i b x^2+x \tanh ^{-1}(1-i d-d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )-\frac {i \text {Subst}\left (\int \frac {\log (1+(-1+i d) x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b}\\ &=\frac {1}{2} i b x^2+x \tanh ^{-1}(1-i d-d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac {i \text {Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}\\ \end {align*}

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Mathematica [A]
time = 20.19, size = 84, normalized size = 0.89 \begin {gather*} x \tanh ^{-1}(1-i d-d \cot (a+b x))-\frac {2 b x \log \left (1+\frac {e^{-2 i (a+b x)}}{-1+i d}\right )+i \text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{i+d}\right )}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[1 - I*d - d*Cot[a + b*x]],x]

[Out]

x*ArcTanh[1 - I*d - d*Cot[a + b*x]] - (2*b*x*Log[1 + 1/((-1 + I*d)*E^((2*I)*(a + b*x)))] + I*PolyLog[2, I/((I
+ d)*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. 364 vs. \(2 (77 ) = 154\).
time = 0.82, size = 365, normalized size = 3.88

method	result	size
derivativedivides	\(-\frac {-\frac {i \arctanh \left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \arctanh \left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d -d \cot \left (b x +a \right )\right )}{2}-\frac {d^{2} \left (\frac {i \dilog \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2 d}+\frac {i \ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2 d}-\frac {i \dilog \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 d +2 i\right )\right )}{2 d +2 i}\right )}{2 d}-\frac {i \ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 d +2 i\right )\right )}{2 d +2 i}\right )}{2 d}-\frac {i \ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4 d}+\frac {i \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right ) \ln \left (i d +d \cot \left (b x +a \right )\right )}{2 d}-\frac {i \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right ) \ln \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2 d}-\frac {i \dilog \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2 d}\right )}{2}}{b d}\)	\(365\)
default	\(-\frac {-\frac {i \arctanh \left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \arctanh \left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d -d \cot \left (b x +a \right )\right )}{2}-\frac {d^{2} \left (\frac {i \dilog \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2 d}+\frac {i \ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2 d}-\frac {i \dilog \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 d +2 i\right )\right )}{2 d +2 i}\right )}{2 d}-\frac {i \ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 d +2 i\right )\right )}{2 d +2 i}\right )}{2 d}-\frac {i \ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4 d}+\frac {i \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right ) \ln \left (i d +d \cot \left (b x +a \right )\right )}{2 d}-\frac {i \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right ) \ln \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2 d}-\frac {i \dilog \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2 d}\right )}{2}}{b d}\)	\(365\)
risch	\(\text {Expression too large to display}\)	\(1623\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. 286 vs. \(2 (67) = 134\).
time = 0.49, size = 286, normalized size = 3.04 \begin {gather*} -\frac {4 \, {\left (b x + a\right )} d {\left (\frac {\log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right )}{d} - \frac {\log \left (i \, \tan \left (b x + a\right ) + 1\right )}{d}\right )} + d {\left (-\frac {2 i \, {\left (\log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (\frac {{\left (d + 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d - 2} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (d + 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d - 2}\right )\right )}}{d} - \frac {2 i \, {\left (\log \left (-\frac {1}{2} \, {\left (d + 2 i\right )} \tan \left (b x + a\right ) + \frac {1}{2} i \, d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\frac {1}{2} \, {\left (d + 2 i\right )} \tan \left (b x + a\right ) - \frac {1}{2} i \, d + 1\right )\right )}}{d} + \frac {2 i \, \log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) - i \, \log \left (i \, \tan \left (b x + a\right ) + 1\right )^{2}}{d} + \frac {2 i \, {\left (\log \left (i \, \tan \left (b x + a\right ) + 1\right ) \log \left (-\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{d}\right )} + 8 \, {\left (b x + a\right )} \operatorname {artanh}\left (i \, d + \frac {d}{\tan \left (b x + a\right )} - 1\right )}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(b*x + a)*d*(log((I*d - 2)*tan(b*x + a) + d)/d - log(I*tan(b*x + a) + 1)/d) + d*(-2*I*(log((I*d - 2)*t
an(b*x + a) + d)*log(((d + 2*I)*tan(b*x + a) - I*d)/(2*I*d - 2) + 1) + dilog(-((d + 2*I)*tan(b*x + a) - I*d)/(
2*I*d - 2)))/d - 2*I*(log(-1/2*(d + 2*I)*tan(b*x + a) + 1/2*I*d)*log(I*tan(b*x + a) + 1) + dilog(1/2*(d + 2*I)
*tan(b*x + a) - 1/2*I*d + 1))/d + (2*I*log((I*d - 2)*tan(b*x + a) + d)*log(I*tan(b*x + a) + 1) - I*log(I*tan(b
*x + a) + 1)^2)/d + 2*I*(log(I*tan(b*x + a) + 1)*log(-1/2*I*tan(b*x + a) + 1/2) + dilog(1/2*I*tan(b*x + a) + 1
/2))/d) + 8*(b*x + a)*arctanh(I*d + d/tan(b*x + a) - 1))/b

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Fricas [A]
time = 0.36, size = 122, normalized size = 1.30 \begin {gather*} \frac {2 i \, b^{2} x^{2} - 2 \, b x \log \left (-\frac {d e^{\left (2 i \, b x + 2 i \, a\right )}}{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 i \, a^{2} - 2 \, {\left (b x + a\right )} \log \left ({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 2 \, a \log \left (\frac {{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}{d + i}\right ) + i \, {\rm Li}_2\left (-{\left (i \, d - 1\right )} 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(-arctanh(-1+I*d+d*cot(b*x+a)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \operatorname {atanh}{\left (d \cot {\left (a + b x \right )} + i d - 1 \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-atanh(-1+I*d+d*cot(b*x+a)),x)

[Out]

-Integral(atanh(d*cot(a + b*x) + I*d - 1), x)

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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(-arctanh(-1+I*d+d*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(-arctanh(d*cot(b*x + a) + I*d - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(d*1i + d*cot(a + b*x) - 1),x)

[Out]

int(-atanh(d*1i + d*cot(a + b*x) - 1), x)

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