3.167 \(\int \frac{\text{sech}^2(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2}{3} i \tanh (x)-\frac{i \text{sech}(x)}{3 (\sinh (x)+i)} \]

[Out]

((-I/3)*Sech[x])/(I + Sinh[x]) - ((2*I)/3)*Tanh[x]

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Rubi [A]  time = 0.0413912, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ -\frac{2}{3} i \tanh (x)-\frac{i \text{sech}(x)}{3 (\sinh (x)+i)} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(I + Sinh[x]),x]

[Out]

((-I/3)*Sech[x])/(I + Sinh[x]) - ((2*I)/3)*Tanh[x]

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0322801, size = 22, normalized size = 0.88 \[ -\frac{1}{3} i \left (2 \tanh (x)+\frac{\text{sech}(x)}{\sinh (x)+i}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^2/(I + Sinh[x]),x]

[Out]

(-I/3)*(Sech[x]/(I + Sinh[x]) + 2*Tanh[x])

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Maple [B]  time = 0.028, size = 49, normalized size = 2. \begin{align*}{-{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}- \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}+{{\frac{2\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{{\frac{3\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(I+sinh(x)),x)

[Out]

-1/2*I/(tanh(1/2*x)-I)-1/(tanh(1/2*x)+I)^2+2/3*I/(tanh(1/2*x)+I)^3-3/2*I/(tanh(1/2*x)+I)

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Maxima [B]  time = 1.10298, size = 72, normalized size = 2.88 \begin{align*} -\frac{8 \, e^{\left (-x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac{4 i}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(I+sinh(x)),x, algorithm="maxima")

[Out]

-8*e^(-x)/(-6*I*e^(-x) - 6*I*e^(-3*x) + 3*e^(-4*x) - 3) + 4*I/(-6*I*e^(-x) - 6*I*e^(-3*x) + 3*e^(-4*x) - 3)

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Fricas [A]  time = 1.63876, size = 76, normalized size = 3.04 \begin{align*} -\frac{8 \, e^{x} + 4 i}{3 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x} - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(I+sinh(x)),x, algorithm="fricas")

[Out]

-(8*e^x + 4*I)/(3*e^(4*x) + 6*I*e^(3*x) + 6*I*e^x - 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{\sinh{\left (x \right )} + i}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(I+sinh(x)),x)

[Out]

Integral(sech(x)**2/(sinh(x) + I), x)

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Giac [A]  time = 1.272, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{2 \,{\left (e^{x} - i\right )}} - \frac{3 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} - 5}{6 \,{\left (e^{x} + i\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(I+sinh(x)),x, algorithm="giac")

[Out]

1/2/(e^x - I) - 1/6*(3*e^(2*x) + 12*I*e^x - 5)/(e^x + I)^3