3.3.17 \(\int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx\) [217]

Optimal. Leaf size=36 \[ -\frac {3}{8} \tanh ^{-1}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x) \]

[Out]

-3/8*arctanh(cosh(x))+1/5*I*coth(x)^5-3/8*coth(x)*csch(x)-1/4*coth(x)^3*csch(x)

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Rubi [A]
time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2785, 2687, 30, 2691, 3855} \begin {gather*} \frac {1}{5} i \coth ^5(x)-\frac {3}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{4} \coth ^3(x) \text {csch}(x)-\frac {3}{8} \coth (x) \text {csch}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^6/(I + Sinh[x]),x]

[Out]

(-3*ArcTanh[Cosh[x]])/8 + (I/5)*Coth[x]^5 - (3*Coth[x]*Csch[x])/8 - (Coth[x]^3*Csch[x])/4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 3855

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

Rubi steps

\begin {align*} \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth ^4(x) \text {csch}^2(x) \, dx\right )+\int \coth ^4(x) \text {csch}(x) \, dx\\ &=-\frac {1}{4} \coth ^3(x) \text {csch}(x)+\frac {3}{4} \int \coth ^2(x) \text {csch}(x) \, dx+\text {Subst}\left (\int x^4 \, dx,x,i \coth (x)\right )\\ &=\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x)+\frac {3}{8} \int \text {csch}(x) \, dx\\ &=-\frac {3}{8} \tanh ^{-1}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x)\\ \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. \(164\) vs. \(2(36)=72\).
time = 0.03, size = 164, normalized size = 4.56 \begin {gather*} \frac {1}{10} i \coth \left (\frac {x}{2}\right )-\frac {5}{32} \text {csch}^2\left (\frac {x}{2}\right )+\frac {7}{160} 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 {1}{160} i \coth \left (\frac {x}{2}\right ) \text {csch}^4\left (\frac {x}{2}\right )+\frac {3}{8} \log \left (\tanh \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 {1}{10} i \tanh \left (\frac {x}{2}\right )-\frac {7}{160} i \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{160} i \text {sech}^4\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^6/(I + Sinh[x]),x]

[Out]

(I/10)*Coth[x/2] - (5*Csch[x/2]^2)/32 + ((7*I)/160)*Coth[x/2]*Csch[x/2]^2 - Csch[x/2]^4/64 + (I/160)*Coth[x/2]
*Csch[x/2]^4 + (3*Log[Tanh[x/2]])/8 - (5*Sech[x/2]^2)/32 + Sech[x/2]^4/64 + (I/10)*Tanh[x/2] - ((7*I)/160)*Sec
h[x/2]^2*Tanh[x/2] + (I/160)*Sech[x/2]^4*Tanh[x/2]

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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. 92 vs. \(2 (27 ) = 54\).
time = 0.76, size = 93, normalized size = 2.58

method	result	size
risch	\(-\frac {-40 i {\mathrm e}^{8 x}+25 \,{\mathrm e}^{9 x}-10 \,{\mathrm e}^{7 x}-80 i {\mathrm e}^{4 x}+10 \,{\mathrm e}^{3 x}-8 i-25 \,{\mathrm e}^{x}}{20 \left ({\mathrm e}^{2 x}-1\right )^{5}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8}\)	\(65\)
default	\(\frac {i \tanh \left (\frac {x}{2}\right )}{16}+\frac {i \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160}+\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}+\frac {i \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{32}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {i}{16 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{160 \tanh \left (\frac {x}{2}\right )^{5}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}+\frac {i}{32 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*I*tanh(1/2*x)+1/160*I*tanh(1/2*x)^5+1/64*tanh(1/2*x)^4+1/32*I*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2+1/16*I/tanh
(1/2*x)-1/8/tanh(1/2*x)^2+1/160*I/tanh(1/2*x)^5+3/8*ln(tanh(1/2*x))+1/32*I/tanh(1/2*x)^3-1/64/tanh(1/2*x)^4

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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. 91 vs. \(2 (26) = 52\).
time = 0.27, size = 91, normalized size = 2.53 \begin {gather*} \frac {25 \, e^{\left (-x\right )} - 10 \, e^{\left (-3 \, x\right )} - 80 i \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-7 \, x\right )} - 40 i \, e^{\left (-8 \, x\right )} - 25 \, e^{\left (-9 \, x\right )} - 8 i}{20 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(25*e^(-x) - 10*e^(-3*x) - 80*I*e^(-4*x) + 10*e^(-7*x) - 40*I*e^(-8*x) - 25*e^(-9*x) - 8*I)/(5*e^(-2*x) -
10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) - 3/8*log(e^(-x) + 1) + 3/8*log(e^(-x) - 1)

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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. 144 vs. \(2 (26) = 52\).
time = 0.36, size = 144, normalized size = 4.00 \begin {gather*} -\frac {15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) - 15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) + 50 \, e^{\left (9 \, x\right )} - 80 i \, e^{\left (8 \, x\right )} - 20 \, e^{\left (7 \, x\right )} - 160 i \, e^{\left (4 \, x\right )} + 20 \, e^{\left (3 \, x\right )} - 50 \, e^{x} - 16 i}{40 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(15*(e^(10*x) - 5*e^(8*x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^(2*x) - 1)*log(e^x + 1) - 15*(e^(10*x) - 5*e^(
8*x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^(2*x) - 1)*log(e^x - 1) + 50*e^(9*x) - 80*I*e^(8*x) - 20*e^(7*x) - 160*I*
e^(4*x) + 20*e^(3*x) - 50*e^x - 16*I)/(e^(10*x) - 5*e^(8*x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^(2*x) - 1)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (36) = 72\).
time = 0.12, size = 100, normalized size = 2.78 \begin {gather*} \frac {3 \log {\left (e^{x} - 1 \right )}}{8} - \frac {3 \log {\left (e^{x} + 1 \right )}}{8} + \frac {- 25 e^{9 x} + 40 i e^{8 x} + 10 e^{7 x} + 80 i e^{4 x} - 10 e^{3 x} + 25 e^{x} + 8 i}{20 e^{10 x} - 100 e^{8 x} + 200 e^{6 x} - 200 e^{4 x} + 100 e^{2 x} - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(exp(x) - 1)/8 - 3*log(exp(x) + 1)/8 + (-25*exp(9*x) + 40*I*exp(8*x) + 10*exp(7*x) + 80*I*exp(4*x) - 10*e
xp(3*x) + 25*exp(x) + 8*I)/(20*exp(10*x) - 100*exp(8*x) + 200*exp(6*x) - 200*exp(4*x) + 100*exp(2*x) - 20)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
time = 0.42, size = 62, normalized size = 1.72 \begin {gather*} -\frac {25 \, e^{\left (9 \, x\right )} - 40 i \, e^{\left (8 \, x\right )} - 10 \, e^{\left (7 \, x\right )} - 80 i \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 25 \, e^{x} - 8 i}{20 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} - \frac {3}{8} \, \log \left (e^{x} + 1\right ) + \frac {3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(25*e^(9*x) - 40*I*e^(8*x) - 10*e^(7*x) - 80*I*e^(4*x) + 10*e^(3*x) - 25*e^x - 8*I)/(e^(2*x) - 1)^5 - 3/
8*log(e^x + 1) + 3/8*log(abs(e^x - 1))

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Mupad [B]
time = 0.97, size = 124, normalized size = 3.44 \begin {gather*} \frac {3\,\ln \left (\frac {3}{4}-\frac {3\,{\mathrm {e}}^x}{4}\right )}{8}-\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 {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {8{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {32{}\mathrm {i}}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^6/(sinh(x) + 1i),x)

[Out]

(3*log(3/4 - (3*exp(x))/4))/8 - (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 + 2i/(exp(2*x) - 1) + 8i/(exp(2*
x) - 1)^2 + 16i/(exp(2*x) - 1)^3 + 16i/(exp(2*x) - 1)^4 + 32i/(5*(exp(2*x) - 1)^5)

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