\(\int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx\) [112]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=-i x-\frac {3}{8} \text {arctanh}(\cosh (x))+\frac {1}{12} \coth ^3(x) (4 i-3 \text {csch}(x))+\frac {1}{8} \coth (x) (8 i-3 \text {csch}(x)) \]

[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] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3966, 3855} \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{8} \text {arctanh}(\cosh (x))-i x+\frac {1}{12} \coth ^3(x) (-3 \text {csch}(x)+4 i)+\frac {1}{8} \coth (x) (-3 \text {csch}(x)+8 i) \]

[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 3855

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

Rule 3966

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 3973

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*} \text {integral}& = \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} \text {arctanh}(\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] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(43)=86\).

Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.26 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=-i x+\frac {2}{3} i \coth \left (\frac {x}{2}\right )-\frac {5}{32} \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{24} i \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^4\left (\frac {x}{2}\right )-\frac {3}{8} \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {3}{8} \log \left (\sinh \left (\frac {x}{2}\right )\right )-\frac {5}{32} \text {sech}^2\left (\frac {x}{2}\right )+\frac {1}{64} \text {sech}^4\left (\frac {x}{2}\right )+\frac {2}{3} i \tanh \left (\frac {x}{2}\right )-\frac {1}{24} i \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \]

[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[Cos
h[x/2]])/8 + (3*Log[Sinh[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]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (34 ) = 68\).

Time = 11.00 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.77

method	result	size
risch	\(-i x -\frac {-48 i {\mathrm e}^{6 x}+15 \,{\mathrm e}^{7 x}+96 i {\mathrm e}^{4 x}+9 \,{\mathrm e}^{5 x}-80 i {\mathrm e}^{2 x}+9 \,{\mathrm e}^{3 x}+32 i+15 \,{\mathrm e}^{x}}{12 \left ({\mathrm e}^{2 x}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8}\)	\(76\)
default	\(\frac {5 i \tanh \left (\frac {x}{2}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )^{4}}{64}+\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {5 i}{8 \tanh \left (\frac {x}{2}\right )}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\)	\(95\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (31) = 62\).

Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 3.65 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=\frac {-24 i \, x e^{\left (8 \, x\right )} - 96 \, {\left (-i \, x - i\right )} e^{\left (6 \, x\right )} - 48 \, {\left (3 i \, x + 4 i\right )} e^{\left (4 \, x\right )} - 32 \, {\left (-3 i \, x - 5 i\right )} e^{\left (2 \, x\right )} - 9 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + 1\right ) + 9 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} - 1\right ) - 24 i \, x - 30 \, e^{\left (7 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 \, e^{\left (3 \, x\right )} - 30 \, e^{x} - 64 i}{24 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]

[In]

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

[Out]

1/24*(-24*I*x*e^(8*x) - 96*(-I*x - I)*e^(6*x) - 48*(3*I*x + 4*I)*e^(4*x) - 32*(-3*I*x - 5*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)

Sympy [F]

\[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth ^{6}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

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

[Out]

Integral(coth(x)**6/(csch(x) + I), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (31) = 62\).

Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.23 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {15 \, e^{\left (-x\right )} + 80 i \, e^{\left (-2 \, x\right )} + 9 \, e^{\left (-3 \, x\right )} - 96 i \, e^{\left (-4 \, x\right )} + 9 \, e^{\left (-5 \, x\right )} + 48 i \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-7 \, x\right )} - 32 i}{12 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {3}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=-i \, x - \frac {15 \, e^{\left (7 \, x\right )} - 48 i \, e^{\left (6 \, x\right )} + 9 \, e^{\left (5 \, x\right )} + 96 i \, e^{\left (4 \, x\right )} + 9 \, e^{\left (3 \, x\right )} - 80 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} + 32 i}{12 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} - \frac {3}{8} \, \log \left (e^{x} + 1\right ) + \frac {3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

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

[Out]

-I*x - 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)/
(e^(2*x) - 1)^4 - 3/8*log(e^x + 1) + 3/8*log(abs(e^x - 1))

Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.47 \[ \int \frac {\coth ^6(x)}{i+\text {csch}(x)} \, dx=\frac {3\,\ln \left (\frac {3}{4}-\frac {3\,{\mathrm {e}}^x}{4}\right )}{8}-x\,1{}\mathrm {i}-\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^x}{4}+\frac {3}{4}\right )}{8}-\frac {5\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {9\,{\mathrm {e}}^x}{2\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {6\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}-\frac {4\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]

[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)