3.112 \(\int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx\)

Optimal. Leaf size=43 \[ -i x-\frac {3}{8} \tanh ^{-1}(\cosh (x))+\frac {1}{12} \coth ^3(x) (-3 \text {csch}(x)+4 i)+\frac {1}{8} \coth (x) (-3 \text {csch}(x)+8 i) \]

[Out]

-I*x-3/8*arctanh(cosh(x))+1/12*coth(x)^3*(4*I-3*csch(x))+1/8*coth(x)*(8*I-3*csch(x))

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ -i x-\frac {3}{8} \tanh ^{-1}(\cosh (x))+\frac {1}{12} \coth ^3(x) (-3 \text {csch}(x)+4 i)+\frac {1}{8} \coth (x) (-3 \text {csch}(x)+8 i) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^6/(I + Csch[x]),x]

[Out]

(-I)*x - (3*ArcTanh[Cosh[x]])/8 + (Coth[x]^3*(4*I - 3*Csch[x]))/12 + (Coth[x]*(8*I - 3*Csch[x]))/8

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx &=\int \coth ^4(x) (-i+\text {csch}(x)) \, dx\\ &=\frac {1}{12} \coth ^3(x) (4 i-3 \text {csch}(x))+\frac {1}{4} \int \coth ^2(x) (-4 i+3 \text {csch}(x)) \, dx\\ &=\frac {1}{12} \coth ^3(x) (4 i-3 \text {csch}(x))+\frac {1}{8} \coth (x) (8 i-3 \text {csch}(x))+\frac {1}{8} \int (-8 i+3 \text {csch}(x)) \, dx\\ &=-i x+\frac {1}{12} \coth ^3(x) (4 i-3 \text {csch}(x))+\frac {1}{8} \coth (x) (8 i-3 \text {csch}(x))+\frac {3}{8} \int \text {csch}(x) \, dx\\ &=-i x-\frac {3}{8} \tanh ^{-1}(\cosh (x))+\frac {1}{12} \coth ^3(x) (4 i-3 \text {csch}(x))+\frac {1}{8} \coth (x) (8 i-3 \text {csch}(x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.04, size = 129, normalized size = 3.00 \[ -i x+\frac {2}{3} i \tanh \left (\frac {x}{2}\right )+\frac {2}{3} i \coth \left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^4\left (\frac {x}{2}\right )-\frac {5}{32} \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{64} \text {sech}^4\left (\frac {x}{2}\right )-\frac {5}{32} \text {sech}^2\left (\frac {x}{2}\right )+\frac {3}{8} \log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{24} i \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{24} i \tanh \left (\frac {x}{2}\right ) \text {sech}^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^6/(I + Csch[x]),x]

[Out]

(-I)*x + ((2*I)/3)*Coth[x/2] - (5*Csch[x/2]^2)/32 + (I/24)*Coth[x/2]*Csch[x/2]^2 - Csch[x/2]^4/64 + (3*Log[Tan
h[x/2]])/8 - (5*Sech[x/2]^2)/32 + Sech[x/2]^4/64 + ((2*I)/3)*Tanh[x/2] - (I/24)*Sech[x/2]^2*Tanh[x/2]

________________________________________________________________________________________

fricas [B]  time = 0.57, size = 154, normalized size = 3.58 \[ \frac {-24 i \, x e^{\left (8 \, x\right )} + {\left (96 i \, x + 96 i\right )} e^{\left (6 \, x\right )} + {\left (-144 i \, x - 192 i\right )} e^{\left (4 \, x\right )} + {\left (96 i \, x + 160 i\right )} e^{\left (2 \, x\right )} - 9 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + 1\right ) + 9 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} - 1\right ) - 24 i \, x - 30 \, e^{\left (7 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 \, e^{\left (3 \, x\right )} - 30 \, e^{x} - 64 i}{24 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="fricas")

[Out]

1/24*(-24*I*x*e^(8*x) + (96*I*x + 96*I)*e^(6*x) + (-144*I*x - 192*I)*e^(4*x) + (96*I*x + 160*I)*e^(2*x) - 9*(e
^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1)*log(e^x + 1) + 9*(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x)
+ 1)*log(e^x - 1) - 24*I*x - 30*e^(7*x) - 18*e^(5*x) - 18*e^(3*x) - 30*e^x - 64*I)/(e^(8*x) - 4*e^(6*x) + 6*e^
(4*x) - 4*e^(2*x) + 1)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 77, normalized size = 1.79 \[ -\frac {15 \, e^{\left (7 \, x\right )} - 48 i \, e^{\left (6 \, x\right )} + 9 \, e^{\left (5 \, x\right )} + 96 i \, e^{\left (4 \, x\right )} + 9 \, e^{\left (3 \, x\right )} - 80 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} + 32 i}{12 \, {\left (i \, e^{\left (2 \, x\right )} - i\right )}^{4}} - i \, \log \left (-i \, e^{x}\right ) - \frac {3}{8} \, \log \left (e^{x} + 1\right ) + \frac {3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="giac")

[Out]

-1/12*(15*e^(7*x) - 48*I*e^(6*x) + 9*e^(5*x) + 96*I*e^(4*x) + 9*e^(3*x) - 80*I*e^(2*x) + 15*e^x + 32*I)/(I*e^(
2*x) - I)^4 - I*log(-I*e^x) - 3/8*log(e^x + 1) + 3/8*log(abs(e^x - 1))

________________________________________________________________________________________

maple [B]  time = 0.25, size = 95, normalized size = 2.21 \[ \frac {5 i \tanh \left (\frac {x}{2}\right )}{8}+\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}+\frac {i \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {5 i}{8 \tanh \left (\frac {x}{2}\right )}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^6/(I+csch(x)),x)

[Out]

5/8*I*tanh(1/2*x)+1/64*tanh(1/2*x)^4+1/24*I*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2+I*ln(tanh(1/2*x)-1)-I*ln(tanh(1/2*
x)+1)-1/64/tanh(1/2*x)^4+5/8*I/tanh(1/2*x)+1/24*I/tanh(1/2*x)^3-1/8/tanh(1/2*x)^2+3/8*ln(tanh(1/2*x))

________________________________________________________________________________________

maxima [B]  time = 0.31, size = 96, normalized size = 2.23 \[ -i \, x + \frac {15 \, e^{\left (-x\right )} + 80 i \, e^{\left (-2 \, x\right )} + 9 \, e^{\left (-3 \, x\right )} - 96 i \, e^{\left (-4 \, x\right )} + 9 \, e^{\left (-5 \, x\right )} + 48 i \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-7 \, x\right )} - 32 i}{12 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {3}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="maxima")

[Out]

-I*x + 1/12*(15*e^(-x) + 80*I*e^(-2*x) + 9*e^(-3*x) - 96*I*e^(-4*x) + 9*e^(-5*x) + 48*I*e^(-6*x) + 15*e^(-7*x)
- 32*I)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 3/8*log(e^(-x) + 1) + 3/8*log(e^(-x) - 1)

________________________________________________________________________________________

mupad [B]  time = 1.86, size = 106, normalized size = 2.47 \[ \frac {3\,\ln \left (\frac {3}{4}-\frac {3\,{\mathrm {e}}^x}{4}\right )}{8}-x\,1{}\mathrm {i}-\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^x}{4}+\frac {3}{4}\right )}{8}-\frac {5\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {9\,{\mathrm {e}}^x}{2\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {6\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}-\frac {4\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^6/(1/sinh(x) + 1i),x)

[Out]

(3*log(3/4 - (3*exp(x))/4))/8 - x*1i - (3*log((3*exp(x))/4 + 3/4))/8 - (5*exp(x))/(4*(exp(2*x) - 1)) - (9*exp(
x))/(2*(exp(2*x) - 1)^2) - (6*exp(x))/(exp(2*x) - 1)^3 - (4*exp(x))/(exp(2*x) - 1)^4 + 4i/(exp(2*x) - 1) + 4i/
(exp(2*x) - 1)^2 + 8i/(3*(exp(2*x) - 1)^3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{6}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**6/(I+csch(x)),x)

[Out]

Integral(coth(x)**6/(csch(x) + I), x)

________________________________________________________________________________________