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

Optimal. Leaf size=29 \[ \frac{1}{5} i \tanh ^5(x)-\frac{1}{3} i \tanh ^3(x)-\frac{1}{5} \text{sech}^5(x) \]

[Out]

-Sech[x]^5/5 - (I/3)*Tanh[x]^3 + (I/5)*Tanh[x]^5

________________________________________________________________________________________

Rubi [A]  time = 0.121423, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3872, 2839, 2606, 30, 2607, 14} \[ \frac{1}{5} i \tanh ^5(x)-\frac{1}{3} i \tanh ^3(x)-\frac{1}{5} \text{sech}^5(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^4/(I + Csch[x]),x]

[Out]

-Sech[x]^5/5 - (I/3)*Tanh[x]^3 + (I/5)*Tanh[x]^5

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2606

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

Rule 30

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

Rule 2607

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\text{sech}^4(x)}{i+\text{csch}(x)} \, dx &=i \int \frac{\text{sech}^3(x) \tanh (x)}{i-\sinh (x)} \, dx\\ &=-\left (i \int \text{sech}^4(x) \tanh ^2(x) \, dx\right )+\int \text{sech}^5(x) \tanh (x) \, dx\\ &=-\operatorname{Subst}\left (\int x^4 \, dx,x,\text{sech}(x)\right )+\operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \tanh (x)\right )\\ &=-\frac{1}{5} \text{sech}^5(x)+\operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \tanh (x)\right )\\ &=-\frac{1}{5} \text{sech}^5(x)-\frac{1}{3} i \tanh ^3(x)+\frac{1}{5} i \tanh ^5(x)\\ \end{align*}

Mathematica [B]  time = 0.104372, size = 96, normalized size = 3.31 \[ \frac{-96 i \sinh (x)+18 i \sinh (2 x)-32 i \sinh (3 x)+9 i \sinh (4 x)+54 \cosh (x)+32 \cosh (2 x)+18 \cosh (3 x)+16 \cosh (4 x)-240}{960 \left (\cosh \left (\frac{x}{2}\right )-i \sinh \left (\frac{x}{2}\right )\right )^3 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^4/(I + Csch[x]),x]

[Out]

(-240 + 54*Cosh[x] + 32*Cosh[2*x] + 18*Cosh[3*x] + 16*Cosh[4*x] - (96*I)*Sinh[x] + (18*I)*Sinh[2*x] - (32*I)*S
inh[3*x] + (9*I)*Sinh[4*x])/(960*(Cosh[x/2] - I*Sinh[x/2])^3*(Cosh[x/2] + I*Sinh[x/2])^5)

________________________________________________________________________________________

Maple [B]  time = 0.045, size = 93, normalized size = 3.2 \begin{align*}{-{\frac{4\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}}+{{\frac{3\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}+{{\frac{2\,i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-5}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-4}- \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}+{{\frac{i}{6}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{{\frac{3\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}}-{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*I/(tanh(1/2*x)-I)^3+3/8*I/(tanh(1/2*x)-I)+2/5*I/(tanh(1/2*x)-I)^5+1/(tanh(1/2*x)-I)^4-1/(tanh(1/2*x)-I)^2
+1/6*I/(tanh(1/2*x)+I)^3-3/8*I/(tanh(1/2*x)+I)-1/4/(tanh(1/2*x)+I)^2

________________________________________________________________________________________

Maxima [B]  time = 0.992296, size = 347, normalized size = 11.97 \begin{align*} \frac{32 \, e^{\left (-x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac{32 i \, e^{\left (-2 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac{96 \, e^{\left (-3 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} - \frac{240 i \, e^{\left (-4 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac{16 i}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32*e^(-x)/(120*I*e^(-x) - 120*e^(-2*x) + 360*I*e^(-3*x) + 360*I*e^(-5*x) + 120*e^(-6*x) + 120*I*e^(-7*x) + 60*
e^(-8*x) - 60) + 32*I*e^(-2*x)/(120*I*e^(-x) - 120*e^(-2*x) + 360*I*e^(-3*x) + 360*I*e^(-5*x) + 120*e^(-6*x) +
120*I*e^(-7*x) + 60*e^(-8*x) - 60) + 96*e^(-3*x)/(120*I*e^(-x) - 120*e^(-2*x) + 360*I*e^(-3*x) + 360*I*e^(-5*
x) + 120*e^(-6*x) + 120*I*e^(-7*x) + 60*e^(-8*x) - 60) - 240*I*e^(-4*x)/(120*I*e^(-x) - 120*e^(-2*x) + 360*I*e
^(-3*x) + 360*I*e^(-5*x) + 120*e^(-6*x) + 120*I*e^(-7*x) + 60*e^(-8*x) - 60) + 16*I/(120*I*e^(-x) - 120*e^(-2*
x) + 360*I*e^(-3*x) + 360*I*e^(-5*x) + 120*e^(-6*x) + 120*I*e^(-7*x) + 60*e^(-8*x) - 60)

________________________________________________________________________________________

Fricas [B]  time = 1.44565, size = 212, normalized size = 7.31 \begin{align*} \frac{60 i \, e^{\left (4 \, x\right )} + 24 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )} + 8 \, e^{x} - 4 i}{15 \, e^{\left (8 \, x\right )} - 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} - 90 i \, e^{\left (5 \, x\right )} - 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} - 30 i \, e^{x} - 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(60*I*e^(4*x) + 24*e^(3*x) - 8*I*e^(2*x) + 8*e^x - 4*I)/(15*e^(8*x) - 30*I*e^(7*x) + 30*e^(6*x) - 90*I*e^(5*x)
- 90*I*e^(3*x) - 30*e^(2*x) - 30*I*e^x - 15)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{4}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(x)**4/(csch(x) + I), x)

________________________________________________________________________________________

Giac [B]  time = 1.16577, size = 74, normalized size = 2.55 \begin{align*} -\frac{-3 i \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5 i}{24 \,{\left (i \, e^{x} - 1\right )}^{3}} + \frac{15 \, e^{\left (4 \, x\right )} - 60 i \, e^{\left (3 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 20 i \, e^{x} + 7}{120 \,{\left (e^{x} - i\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(-3*I*e^(2*x) + 12*e^x + 5*I)/(I*e^x - 1)^3 + 1/120*(15*e^(4*x) - 60*I*e^(3*x) - 10*e^(2*x) + 20*I*e^x +
7)/(e^x - I)^5