3.2.90 \(\int \cot ^{-1}(c+d \tanh (a+b x)) \, dx\) [190]

Optimal. Leaf size=174 \[ x \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{2} i x \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{2} i x \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i \text {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b} \]

[Out]

x*arccot(c+d*tanh(b*x+a))-1/2*I*x*ln(1+(I-c-d)*exp(2*b*x+2*a)/(I-c+d))+1/2*I*x*ln(1+(I+c+d)*exp(2*b*x+2*a)/(I+
c-d))-1/4*I*polylog(2,-(I-c-d)*exp(2*b*x+2*a)/(I-c+d))/b+1/4*I*polylog(2,-(I+c+d)*exp(2*b*x+2*a)/(I+c-d))/b

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Rubi [A]
time = 0.16, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5300, 2221, 2317, 2438} \begin {gather*} -\frac {i \text {Li}_2\left (-\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{4 b}+\frac {i \text {Li}_2\left (-\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{4 b}-\frac {1}{2} i x \log \left (1+\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )+\frac {1}{2} i x \log \left (1+\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )+x \cot ^{-1}(d \tanh (a+b x)+c) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[c + d*Tanh[a + b*x]],x]

[Out]

x*ArcCot[c + d*Tanh[a + b*x]] - (I/2)*x*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/2)*x*Log[1 + (
(I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b
+ ((I/4)*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b

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 5300

Int[ArcCot[(c_.) + (d_.)*Tanh[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*ArcCot[c + d*Tanh[a + b*x]], x] + (-Di
st[I*b*(I - c - d), Int[x*(E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x))), x], x] + Dist[I*b*(I +
c + d), Int[x*(E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x))), x], x]) /; FreeQ[{a, b, c, d}, x] &
& NeQ[(c - d)^2, -1]

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(363\) vs. \(2(174)=348\).
time = 0.84, size = 363, normalized size = 2.09 \begin {gather*} x \cot ^{-1}(c+d \tanh (a+b x))-\frac {i \left (-2 i a \text {ArcTan}\left (\frac {1+c^2-d^2+\left (1+c^2+2 c d+d^2\right ) e^{2 (a+b x)}}{2 d}\right )+(a+b x) \log \left (1-\frac {\sqrt {-i+c+d} e^{a+b x}}{\sqrt {i-c+d}}\right )+(a+b x) \log \left (1+\frac {\sqrt {-i+c+d} e^{a+b x}}{\sqrt {i-c+d}}\right )-(a+b x) \log \left (1-\frac {\sqrt {i+c+d} e^{a+b x}}{\sqrt {-i-c+d}}\right )-(a+b x) \log \left (1+\frac {\sqrt {i+c+d} e^{a+b x}}{\sqrt {-i-c+d}}\right )+\text {PolyLog}\left (2,-\frac {\sqrt {-i+c+d} e^{a+b x}}{\sqrt {i-c+d}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {-i+c+d} e^{a+b x}}{\sqrt {i-c+d}}\right )-\text {PolyLog}\left (2,-\frac {\sqrt {i+c+d} e^{a+b x}}{\sqrt {-i-c+d}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {i+c+d} e^{a+b x}}{\sqrt {-i-c+d}}\right )\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[c + d*Tanh[a + b*x]],x]

[Out]

x*ArcCot[c + d*Tanh[a + b*x]] - ((I/2)*((-2*I)*a*ArcTan[(1 + c^2 - d^2 + (1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x
)))/(2*d)] + (a + b*x)*Log[1 - (Sqrt[-I + c + d]*E^(a + b*x))/Sqrt[I - c + d]] + (a + b*x)*Log[1 + (Sqrt[-I +
c + d]*E^(a + b*x))/Sqrt[I - c + d]] - (a + b*x)*Log[1 - (Sqrt[I + c + d]*E^(a + b*x))/Sqrt[-I - c + d]] - (a
+ b*x)*Log[1 + (Sqrt[I + c + d]*E^(a + b*x))/Sqrt[-I - c + d]] + PolyLog[2, -((Sqrt[-I + c + d]*E^(a + b*x))/S
qrt[I - c + d])] + PolyLog[2, (Sqrt[-I + c + d]*E^(a + b*x))/Sqrt[I - c + d]] - PolyLog[2, -((Sqrt[I + c + d]*
E^(a + b*x))/Sqrt[-I - c + d])] - PolyLog[2, (Sqrt[I + c + d]*E^(a + b*x))/Sqrt[-I - c + d]]))/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. \(2 (150 ) = 300\).
time = 0.94, size = 365, normalized size = 2.10

method	result	size
derivativedivides	\(\frac {-\frac {\mathrm {arccot}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\mathrm {arccot}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {i \ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c +d}\right )}{2 d}-\frac {i \ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c -d}\right )}{2 d}+\frac {i \dilog \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c +d}\right )}{2 d}-\frac {i \dilog \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c -d}\right )}{2 d}-\frac {i \ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c -d}\right )}{2 d}+\frac {i \ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c +d}\right )}{2 d}-\frac {i \dilog \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c -d}\right )}{2 d}+\frac {i \dilog \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c +d}\right )}{2 d}\right )}{2}}{b d}\)	\(365\)
default	\(\frac {-\frac {\mathrm {arccot}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\mathrm {arccot}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {i \ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c +d}\right )}{2 d}-\frac {i \ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c -d}\right )}{2 d}+\frac {i \dilog \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c +d}\right )}{2 d}-\frac {i \dilog \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c -d}\right )}{2 d}-\frac {i \ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c -d}\right )}{2 d}+\frac {i \ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c +d}\right )}{2 d}-\frac {i \dilog \left (\frac {i+d \tanh \left (b x +a \right )+c}{i+c -d}\right )}{2 d}+\frac {i \dilog \left (\frac {i-d \tanh \left (b x +a \right )-c}{i-c +d}\right )}{2 d}\right )}{2}}{b d}\)	\(365\)
risch	\(\text {Expression too large to display}\)	\(4171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(c+d*tanh(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/d*(-1/2*arccot(c+d*tanh(b*x+a))*d*ln(-d*tanh(b*x+a)+d)+1/2*arccot(c+d*tanh(b*x+a))*d*ln(-d*tanh(b*x+a)-d)-
1/2*d^2*(1/2*I/d*ln(-d*tanh(b*x+a)+d)*ln((I+d*tanh(b*x+a)+c)/(I+c+d))-1/2*I/d*ln(-d*tanh(b*x+a)+d)*ln((I-d*tan
h(b*x+a)-c)/(I-c-d))+1/2*I/d*dilog((I+d*tanh(b*x+a)+c)/(I+c+d))-1/2*I/d*dilog((I-d*tanh(b*x+a)-c)/(I-c-d))-1/2
*I/d*ln(-d*tanh(b*x+a)-d)*ln((I+d*tanh(b*x+a)+c)/(I+c-d))+1/2*I/d*ln(-d*tanh(b*x+a)-d)*ln((I-d*tanh(b*x+a)-c)/
(I-c+d))-1/2*I/d*dilog((I+d*tanh(b*x+a)+c)/(I+c-d))+1/2*I/d*dilog((I-d*tanh(b*x+a)-c)/(I-c+d))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c+d*tanh(b*x+a)),x, algorithm="maxima")

[Out]

4*b*d*integrate(x*e^(2*b*x + 2*a)/(c^2 - 2*c*d + d^2 + (c^2*e^(4*a) + 2*c*d*e^(4*a) + d^2*e^(4*a) + e^(4*a))*e
^(4*b*x) + 2*(c^2*e^(2*a) - d^2*e^(2*a) + e^(2*a))*e^(2*b*x) + 1), x) + x*arctan2(e^(2*b*x + 2*a) + 1, (c*e^(2
*a) + d*e^(2*a))*e^(2*b*x) + c - d)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 825 vs. \(2 (128) = 256\).
time = 1.70, size = 825, normalized size = 4.74 \begin {gather*} \frac {2 \, b x \arctan \left (\frac {\cosh \left (b x + a\right )}{c \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) + i \, a \log \left (2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c^{2} - d^{2} - 2 i \, d + 1\right )} \sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}}\right ) + i \, a \log \left (2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c^{2} - d^{2} - 2 i \, d + 1\right )} \sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}}\right ) - i \, a \log \left (2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c^{2} - d^{2} + 2 i \, d + 1\right )} \sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}}\right ) - i \, a \log \left (2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c^{2} + 2 \, c d + d^{2} + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c^{2} - d^{2} + 2 i \, d + 1\right )} \sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}}\right ) + {\left (-i \, b x - i \, a\right )} \log \left (\sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b x - i \, a\right )} \log \left (-\sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (\sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (-\sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - i \, {\rm Li}_2\left (\sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - i \, {\rm Li}_2\left (-\sqrt {-\frac {c^{2} - d^{2} + 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + i \, {\rm Li}_2\left (\sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + i \, {\rm Li}_2\left (-\sqrt {-\frac {c^{2} - d^{2} - 2 i \, d + 1}{c^{2} - 2 \, c d + d^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c+d*tanh(b*x+a)),x, algorithm="fricas")

[Out]

1/2*(2*b*x*arctan(cosh(b*x + a)/(c*cosh(b*x + a) + d*sinh(b*x + a))) + I*a*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(
b*x + a) + 2*(c^2 + 2*c*d + d^2 + 1)*sinh(b*x + a) + 2*(c^2 - d^2 - 2*I*d + 1)*sqrt(-(c^2 - d^2 + 2*I*d + 1)/(
c^2 - 2*c*d + d^2 + 1))) + I*a*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^2 + 2*c*d + d^2 + 1)*sinh(b*
x + a) - 2*(c^2 - d^2 - 2*I*d + 1)*sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))) - I*a*log(2*(c^2 +
2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^2 + 2*c*d + d^2 + 1)*sinh(b*x + a) + 2*(c^2 - d^2 + 2*I*d + 1)*sqrt(-(c^
2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))) - I*a*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^2 + 2*
c*d + d^2 + 1)*sinh(b*x + a) - 2*(c^2 - d^2 + 2*I*d + 1)*sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1)
)) + (-I*b*x - I*a)*log(sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))
+ 1) + (-I*b*x - I*a)*log(-sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x +
a)) + 1) + (I*b*x + I*a)*log(sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x
+ a)) + 1) + (I*b*x + I*a)*log(-sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(
b*x + a)) + 1) - I*dilog(sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a)
)) - I*dilog(-sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) + I*dilo
g(sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) + I*dilog(-sqrt(-(c^
2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {acot}{\left (c + d \tanh {\left (a + b x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(c+d*tanh(b*x+a)),x)

[Out]

Integral(acot(c + d*tanh(a + b*x)), 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(arccot(c+d*tanh(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccot(d*tanh(b*x + a) + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(c + d*tanh(a + b*x)),x)

[Out]

int(acot(c + d*tanh(a + b*x)), x)

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