3.47 \(\int \frac{\text{csch}^4(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=47 \[ \frac{4}{3} i \coth ^3(x)-4 i \coth (x)+\frac{3}{2} \tanh ^{-1}(\cosh (x))-\frac{3}{2} \coth (x) \text{csch}(x)+\frac{\coth (x) \text{csch}^2(x)}{\sinh (x)+i} \]

[Out]

(3*ArcTanh[Cosh[x]])/2 - (4*I)*Coth[x] + ((4*I)/3)*Coth[x]^3 - (3*Coth[x]*Csch[x])/2 + (Coth[x]*Csch[x]^2)/(I
+ Sinh[x])

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Rubi [A]  time = 0.0726019, antiderivative size = 47, normalized size of antiderivative = 1., 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 = {2768, 2748, 3767, 3768, 3770} \[ \frac{4}{3} i \coth ^3(x)-4 i \coth (x)+\frac{3}{2} \tanh ^{-1}(\cosh (x))-\frac{3}{2} \coth (x) \text{csch}(x)+\frac{\coth (x) \text{csch}^2(x)}{\sinh (x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(I + Sinh[x]),x]

[Out]

(3*ArcTanh[Cosh[x]])/2 - (4*I)*Coth[x] + ((4*I)/3)*Coth[x]^3 - (3*Coth[x]*Csch[x])/2 + (Coth[x]*Csch[x]^2)/(I
+ Sinh[x])

Rule 2768

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b
^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x])), x] + Dist[d/(a*(b*c - a*
d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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{\text{csch}^4(x)}{i+\sinh (x)} \, dx &=\frac{\coth (x) \text{csch}^2(x)}{i+\sinh (x)}+\int \text{csch}^4(x) (-4 i+3 \sinh (x)) \, dx\\ &=\frac{\coth (x) \text{csch}^2(x)}{i+\sinh (x)}-4 i \int \text{csch}^4(x) \, dx+3 \int \text{csch}^3(x) \, dx\\ &=-\frac{3}{2} \coth (x) \text{csch}(x)+\frac{\coth (x) \text{csch}^2(x)}{i+\sinh (x)}-\frac{3}{2} \int \text{csch}(x) \, dx+4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )\\ &=\frac{3}{2} \tanh ^{-1}(\cosh (x))-4 i \coth (x)+\frac{4}{3} i \coth ^3(x)-\frac{3}{2} \coth (x) \text{csch}(x)+\frac{\coth (x) \text{csch}^2(x)}{i+\sinh (x)}\\ \end{align*}

Mathematica [A]  time = 0.21572, size = 53, normalized size = 1.13 \[ \frac{1}{6} \text{sech}(x) \left (-16 i \sinh (x)+2 i \text{csch}^3(x)-3 \text{csch}^2(x)-8 i \text{csch}(x)+9 \sqrt{\cosh ^2(x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(x)}\right )-9\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(I + Sinh[x]),x]

[Out]

(Sech[x]*(-9 + 9*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2] - (8*I)*Csch[x] - 3*Csch[x]^2 + (2*I)*Csch[x]^3 - (1
6*I)*Sinh[x]))/6

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Maple [A]  time = 0.035, size = 71, normalized size = 1.5 \begin{align*} -{\frac{7\,i}{8}}\tanh \left ({\frac{x}{2}} \right ) +{\frac{i}{24}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{{\frac{i}{24}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{{\frac{7\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(I+sinh(x)),x)

[Out]

-7/8*I*tanh(1/2*x)+1/24*I*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2+1/24*I/tanh(1/2*x)^3-7/8*I/tanh(1/2*x)-1/8/tanh(1/2*
x)^2-3/2*ln(tanh(1/2*x))-2*I/(tanh(1/2*x)+I)

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Maxima [B]  time = 1.18324, size = 142, normalized size = 3.02 \begin{align*} \frac{16 \,{\left (-7 i \, e^{\left (-x\right )} + 39 \, e^{\left (-2 \, x\right )} + 24 i \, e^{\left (-3 \, x\right )} - 24 \, e^{\left (-4 \, x\right )} - 9 i \, e^{\left (-5 \, x\right )} + 9 \, e^{\left (-6 \, x\right )} - 16\right )}}{48 \, e^{\left (-x\right )} + 144 i \, e^{\left (-2 \, x\right )} - 144 \, e^{\left (-3 \, x\right )} - 144 i \, e^{\left (-4 \, x\right )} + 144 \, e^{\left (-5 \, x\right )} + 48 i \, e^{\left (-6 \, x\right )} - 48 \, e^{\left (-7 \, x\right )} - 48 i} + \frac{3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac{3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x)),x, algorithm="maxima")

[Out]

16*(-7*I*e^(-x) + 39*e^(-2*x) + 24*I*e^(-3*x) - 24*e^(-4*x) - 9*I*e^(-5*x) + 9*e^(-6*x) - 16)/(48*e^(-x) + 144
*I*e^(-2*x) - 144*e^(-3*x) - 144*I*e^(-4*x) + 144*e^(-5*x) + 48*I*e^(-6*x) - 48*e^(-7*x) - 48*I) + 3/2*log(e^(
-x) + 1) - 3/2*log(e^(-x) - 1)

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Fricas [B]  time = 2.05862, size = 545, normalized size = 11.6 \begin{align*} \frac{{\left (9 \, e^{\left (7 \, x\right )} + 9 i \, e^{\left (6 \, x\right )} - 27 \, e^{\left (5 \, x\right )} - 27 i \, e^{\left (4 \, x\right )} + 27 \, e^{\left (3 \, x\right )} + 27 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 9 i\right )} \log \left (e^{x} + 1\right ) -{\left (9 \, e^{\left (7 \, x\right )} + 9 i \, e^{\left (6 \, x\right )} - 27 \, e^{\left (5 \, x\right )} - 27 i \, e^{\left (4 \, x\right )} + 27 \, e^{\left (3 \, x\right )} + 27 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 9 i\right )} \log \left (e^{x} - 1\right ) - 18 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} + 48 \, e^{\left (4 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 78 \, e^{\left (2 \, x\right )} - 14 i \, e^{x} + 32}{6 \, e^{\left (7 \, x\right )} + 6 i \, e^{\left (6 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 i \, e^{\left (4 \, x\right )} + 18 \, e^{\left (3 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 6 i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x)),x, algorithm="fricas")

[Out]

((9*e^(7*x) + 9*I*e^(6*x) - 27*e^(5*x) - 27*I*e^(4*x) + 27*e^(3*x) + 27*I*e^(2*x) - 9*e^x - 9*I)*log(e^x + 1)
- (9*e^(7*x) + 9*I*e^(6*x) - 27*e^(5*x) - 27*I*e^(4*x) + 27*e^(3*x) + 27*I*e^(2*x) - 9*e^x - 9*I)*log(e^x - 1)
- 18*e^(6*x) - 18*I*e^(5*x) + 48*e^(4*x) + 48*I*e^(3*x) - 78*e^(2*x) - 14*I*e^x + 32)/(6*e^(7*x) + 6*I*e^(6*x
) - 18*e^(5*x) - 18*I*e^(4*x) + 18*e^(3*x) + 18*I*e^(2*x) - 6*e^x - 6*I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**4/(I+sinh(x)),x)

[Out]

Timed out

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Giac [A]  time = 1.25883, size = 78, normalized size = 1.66 \begin{align*} -\frac{2}{e^{x} + i} - \frac{3 \, e^{\left (5 \, x\right )} + 6 i \, e^{\left (4 \, x\right )} - 24 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 10 i}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + \frac{3}{2} \, \log \left (e^{x} + 1\right ) - \frac{3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x)),x, algorithm="giac")

[Out]

-2/(e^x + I) - 1/3*(3*e^(5*x) + 6*I*e^(4*x) - 24*I*e^(2*x) - 3*e^x + 10*I)/(e^(2*x) - 1)^3 + 3/2*log(e^x + 1)
- 3/2*log(abs(e^x - 1))