3.3.5 \(\int x^2 \cot ^{-1}(c+d \coth (a+b x)) \, dx\) [205]

Optimal. Leaf size=351 \[ \frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x^2 \text {PolyLog}\left (2,\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x^2 \text {PolyLog}\left (2,\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i x \text {PolyLog}\left (3,\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b^2}-\frac {i x \text {PolyLog}\left (3,\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b^2}-\frac {i \text {PolyLog}\left (4,\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{8 b^3}+\frac {i \text {PolyLog}\left (4,\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{8 b^3} \]

[Out]

1/3*x^3*arccot(c+d*coth(b*x+a))-1/6*I*x^3*ln(1-(I-c-d)*exp(2*b*x+2*a)/(I-c+d))+1/6*I*x^3*ln(1-(I+c+d)*exp(2*b*
x+2*a)/(I+c-d))-1/4*I*x^2*polylog(2,(I-c-d)*exp(2*b*x+2*a)/(I-c+d))/b+1/4*I*x^2*polylog(2,(I+c+d)*exp(2*b*x+2*
a)/(I+c-d))/b+1/4*I*x*polylog(3,(I-c-d)*exp(2*b*x+2*a)/(I-c+d))/b^2-1/4*I*x*polylog(3,(I+c+d)*exp(2*b*x+2*a)/(
I+c-d))/b^2-1/8*I*polylog(4,(I-c-d)*exp(2*b*x+2*a)/(I-c+d))/b^3+1/8*I*polylog(4,(I+c+d)*exp(2*b*x+2*a)/(I+c-d)
)/b^3

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Rubi [A]
time = 0.34, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5310, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {i \text {Li}_4\left (\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{8 b^3}+\frac {i \text {Li}_4\left (\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{8 b^3}+\frac {i x \text {Li}_3\left (\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{4 b^2}-\frac {i x \text {Li}_3\left (\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{4 b^2}-\frac {i x^2 \text {Li}_2\left (\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{4 b}+\frac {i x^2 \text {Li}_2\left (\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{4 b}-\frac {1}{6} i x^3 \log \left (1-\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )+\frac {1}{3} x^3 \cot ^{-1}(d \coth (a+b x)+c) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCot[c + d*Coth[a + b*x]],x]

[Out]

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

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 5310

Int[ArcCot[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*(ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1))), x] + (Dist[I*b*((I - c - d)/(f*(m + 1))), Int[(e + f*x)^(m + 1
)*(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)/(f*(m + 1))), In
t[(e + f*x)^(m + 1)*(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,
e, f}, x] && IGtQ[m, 0] && NeQ[(c - 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]

Rule 6744

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

Rubi steps

\begin {align*} \int x^2 \cot ^{-1}(c+d \coth (a+b x)) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))+\frac {1}{3} (b (1-i (c+d))) \int \frac {e^{2 a+2 b x} x^3}{i+c-d+(-i-c-d) e^{2 a+2 b x}} \, dx-\frac {1}{3} (b (1+i (c+d))) \int \frac {e^{2 a+2 b x} x^3}{i-c+d+(-i+c+d) e^{2 a+2 b x}} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {1}{2} i \int x^2 \log \left (1+\frac {(-i-c-d) e^{2 a+2 b x}}{i+c-d}\right ) \, dx+\frac {1}{2} i \int x^2 \log \left (1+\frac {(-i+c+d) e^{2 a+2 b x}}{i-c+d}\right ) \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x^2 \text {Li}_2\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x^2 \text {Li}_2\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}-\frac {i \int x \text {Li}_2\left (-\frac {(-i-c-d) e^{2 a+2 b x}}{i+c-d}\right ) \, dx}{2 b}+\frac {i \int x \text {Li}_2\left (-\frac {(-i+c+d) e^{2 a+2 b x}}{i-c+d}\right ) \, dx}{2 b}\\ &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x^2 \text {Li}_2\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x^2 \text {Li}_2\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i x \text {Li}_3\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b^2}-\frac {i x \text {Li}_3\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b^2}+\frac {i \int \text {Li}_3\left (-\frac {(-i-c-d) e^{2 a+2 b x}}{i+c-d}\right ) \, dx}{4 b^2}-\frac {i \int \text {Li}_3\left (-\frac {(-i+c+d) e^{2 a+2 b x}}{i-c+d}\right ) \, dx}{4 b^2}\\ &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x^2 \text {Li}_2\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x^2 \text {Li}_2\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i x \text {Li}_3\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b^2}-\frac {i x \text {Li}_3\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b^2}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {(-i+c+d) x}{-i+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {(i+c+d) x}{i+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}\\ &=\frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {1}{6} i x^3 \log \left (1-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x^2 \text {Li}_2\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x^2 \text {Li}_2\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i x \text {Li}_3\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b^2}-\frac {i x \text {Li}_3\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b^2}-\frac {i \text {Li}_4\left (\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{8 b^3}+\frac {i \text {Li}_4\left (\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{8 b^3}\\ \end {align*}

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Mathematica [A]
time = 3.40, size = 299, normalized size = 0.85 \begin {gather*} \frac {1}{3} x^3 \cot ^{-1}(c+d \coth (a+b x))-\frac {i \left (4 b^3 x^3 \log \left (1+\frac {(-i+c+d) e^{2 (a+b x)}}{i-c+d}\right )-4 b^3 x^3 \log \left (1+\frac {(i+c+d) e^{2 (a+b x)}}{-i-c+d}\right )+6 b^2 x^2 \text {PolyLog}\left (2,\frac {(-i+c+d) e^{2 (a+b x)}}{-i+c-d}\right )-6 b^2 x^2 \text {PolyLog}\left (2,\frac {(i+c+d) e^{2 (a+b x)}}{i+c-d}\right )-6 b x \text {PolyLog}\left (3,\frac {(-i+c+d) e^{2 (a+b x)}}{-i+c-d}\right )+6 b x \text {PolyLog}\left (3,\frac {(i+c+d) e^{2 (a+b x)}}{i+c-d}\right )+3 \text {PolyLog}\left (4,\frac {(-i+c+d) e^{2 (a+b x)}}{-i+c-d}\right )-3 \text {PolyLog}\left (4,\frac {(i+c+d) e^{2 (a+b x)}}{i+c-d}\right )\right )}{24 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCot[c + d*Coth[a + b*x]],x]

[Out]

(x^3*ArcCot[c + d*Coth[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(I - c + d)] -
4*b^3*x^3*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(-I - c + d)] + 6*b^2*x^2*PolyLog[2, ((-I + c + d)*E^(2*(a +
b*x)))/(-I + c - d)] - 6*b^2*x^2*PolyLog[2, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] - 6*b*x*PolyLog[3, ((-I
+ c + d)*E^(2*(a + b*x)))/(-I + c - d)] + 6*b*x*PolyLog[3, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 3*Pol
yLog[4, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 3*PolyLog[4, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)]
))/b^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 26.91, size = 6888, normalized size = 19.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(6888\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccot(c+d*coth(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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(x^2*arccot(c+d*coth(b*x+a)),x, algorithm="maxima")

[Out]

1/3*x^3*arctan2(e^(2*b*x + 2*a) - 1, (c*e^(2*a) + d*e^(2*a))*e^(2*b*x) - c + d) - 4*b*d*integrate(1/3*x^3*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)

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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. 1269 vs. \(2 (259) = 518\).
time = 2.26, size = 1269, normalized size = 3.62 \begin {gather*} \frac {2 \, b^{3} x^{3} \arctan \left (\frac {\sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + c \sinh \left (b x + a\right )}\right ) - 3 i \, b^{2} x^{2} {\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 ) - 3 i \, b^{2} x^{2} {\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 ) + 3 i \, b^{2} x^{2} {\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 ) + 3 i \, b^{2} x^{2} {\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 \, a^{3} \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^{3} \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^{3} \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^{3} \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 ) + 6 i \, b x {\rm polylog}\left (3, \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 ) + 6 i \, b x {\rm polylog}\left (3, -\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 ) - 6 i \, b x {\rm polylog}\left (3, \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 ) - 6 i \, b x {\rm polylog}\left (3, -\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 ) + {\left (-i \, b^{3} x^{3} - i \, a^{3}\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^{3} x^{3} - i \, a^{3}\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^{3} x^{3} + i \, a^{3}\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^{3} x^{3} + i \, a^{3}\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 ) - 6 i \, {\rm polylog}\left (4, \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 ) - 6 i \, {\rm polylog}\left (4, -\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 ) + 6 i \, {\rm polylog}\left (4, \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 ) + 6 i \, {\rm polylog}\left (4, -\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 )}{6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccot(c+d*coth(b*x+a)),x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^3*arctan(sinh(b*x + a)/(d*cosh(b*x + a) + c*sinh(b*x + a))) - 3*I*b^2*x^2*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))) - 3*I*b^2*x^2*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))) + 3*I*b^2*x^2*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))) + 3*I*b^2*x^2*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*a^3*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^3*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^3*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^3*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)))
+ 6*I*b*x*polylog(3, 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))) +
6*I*b*x*polylog(3, -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))) -
6*I*b*x*polylog(3, 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))) - 6*
I*b*x*polylog(3, -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
*b^3*x^3 - I*a^3)*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^3*x^3 - I*a^3)*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^3*x^3 + I*a^3)*log(sqrt((c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + s
inh(b*x + a)) + 1) + (I*b^3*x^3 + I*a^3)*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) - 6*I*polylog(4, 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))) - 6*I*polylog(4, -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))) + 6*I*polylog(4, 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))) + 6*I*polylog(4, -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^3

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acot(c+d*coth(b*x+a)),x)

[Out]

Timed out

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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^2*arccot(c+d*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(x^2*arccot(d*coth(b*x + a) + c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\mathrm {acot}\left (c+d\,\mathrm {coth}\left (a+b\,x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*acot(c + d*coth(a + b*x)),x)

[Out]

int(x^2*acot(c + d*coth(a + b*x)), x)

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