3.180 \(\int \frac{\text{sech}^4(x)}{(i+\sinh (x))^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{4 \tanh ^3(x)}{21}-\frac{4 \tanh (x)}{7}-\frac{\text{sech}^3(x)}{7 (\sinh (x)+i)}-\frac{i \text{sech}^3(x)}{7 (\sinh (x)+i)^2} \]

[Out]

((-I/7)*Sech[x]^3)/(I + Sinh[x])^2 - Sech[x]^3/(7*(I + Sinh[x])) - (4*Tanh[x])/7 + (4*Tanh[x]^3)/21

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Rubi [A]  time = 0.0784621, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2672, 3767} \[ \frac{4 \tanh ^3(x)}{21}-\frac{4 \tanh (x)}{7}-\frac{\text{sech}^3(x)}{7 (\sinh (x)+i)}-\frac{i \text{sech}^3(x)}{7 (\sinh (x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^4/(I + Sinh[x])^2,x]

[Out]

((-I/7)*Sech[x]^3)/(I + Sinh[x])^2 - Sech[x]^3/(7*(I + Sinh[x])) - (4*Tanh[x])/7 + (4*Tanh[x]^3)/21

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]

Rubi steps

\begin{align*} \int \frac{\text{sech}^4(x)}{(i+\sinh (x))^2} \, dx &=-\frac{i \text{sech}^3(x)}{7 (i+\sinh (x))^2}-\frac{5}{7} i \int \frac{\text{sech}^4(x)}{i+\sinh (x)} \, dx\\ &=-\frac{i \text{sech}^3(x)}{7 (i+\sinh (x))^2}-\frac{\text{sech}^3(x)}{7 (i+\sinh (x))}-\frac{4}{7} \int \text{sech}^4(x) \, dx\\ &=-\frac{i \text{sech}^3(x)}{7 (i+\sinh (x))^2}-\frac{\text{sech}^3(x)}{7 (i+\sinh (x))}-\frac{4}{7} i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )\\ &=-\frac{i \text{sech}^3(x)}{7 (i+\sinh (x))^2}-\frac{\text{sech}^3(x)}{7 (i+\sinh (x))}-\frac{4 \tanh (x)}{7}+\frac{4 \tanh ^3(x)}{21}\\ \end{align*}

Mathematica [A]  time = 0.03553, size = 47, normalized size = 0.96 \[ -\frac{\text{sech}^3(x) (-14 \sinh (x)-3 \sinh (3 x)+\sinh (5 x)+8 i \cosh (2 x)+4 i \cosh (4 x))}{42 (\sinh (x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^4/(I + Sinh[x])^2,x]

[Out]

-(Sech[x]^3*((8*I)*Cosh[2*x] + (4*I)*Cosh[4*x] - 14*Sinh[x] - 3*Sinh[3*x] + Sinh[5*x]))/(42*(I + Sinh[x])^2)

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Maple [B]  time = 0.052, size = 116, normalized size = 2.4 \begin{align*}{-{\frac{i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+{\frac{1}{12} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}}-{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}+{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-6}}-{5\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-4}}+{{\frac{23\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}+{\frac{4}{7} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-7}}-4\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-5}+{\frac{55}{12} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{\frac{13}{8} \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)^4/(I+sinh(x))^2,x)

[Out]

-1/8*I/(tanh(1/2*x)-I)^2+1/12/(tanh(1/2*x)-I)^3-3/8/(tanh(1/2*x)-I)+2*I/(tanh(1/2*x)+I)^6-5*I/(tanh(1/2*x)+I)^
4+23/8*I/(tanh(1/2*x)+I)^2+4/7/(tanh(1/2*x)+I)^7-4/(tanh(1/2*x)+I)^5+55/12/(tanh(1/2*x)+I)^3-13/8/(tanh(1/2*x)
+I)

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Maxima [B]  time = 1.16245, size = 428, normalized size = 8.73 \begin{align*} -\frac{64 i \, e^{\left (-x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac{48 \, e^{\left (-2 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac{128 i \, e^{\left (-3 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac{224 \, e^{\left (-4 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac{16}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

-64*I*e^(-x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e
^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) + 21) + 48*e^(-2*x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e
^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) + 21) - 128*I*e^(-3*x)/(8
4*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*
e^(-9*x) + 21*e^(-10*x) + 21) + 224*e^(-4*x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e^(-4*x) - 294*
e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) + 21) - 16/(84*I*e^(-x) - 63*e^(-2*x) +
168*I*e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) +
21)

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Fricas [B]  time = 1.76329, size = 259, normalized size = 5.29 \begin{align*} -\frac{224 \, e^{\left (4 \, x\right )} + 128 i \, e^{\left (3 \, x\right )} + 48 \, e^{\left (2 \, x\right )} + 64 i \, e^{x} - 16}{21 \, e^{\left (10 \, x\right )} + 84 i \, e^{\left (9 \, x\right )} - 63 \, e^{\left (8 \, x\right )} + 168 i \, e^{\left (7 \, x\right )} - 294 \, e^{\left (6 \, x\right )} - 294 \, e^{\left (4 \, x\right )} - 168 i \, e^{\left (3 \, x\right )} - 63 \, e^{\left (2 \, x\right )} - 84 i \, e^{x} + 21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

-(224*e^(4*x) + 128*I*e^(3*x) + 48*e^(2*x) + 64*I*e^x - 16)/(21*e^(10*x) + 84*I*e^(9*x) - 63*e^(8*x) + 168*I*e
^(7*x) - 294*e^(6*x) - 294*e^(4*x) - 168*I*e^(3*x) - 63*e^(2*x) - 84*I*e^x + 21)

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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(sech(x)**4/(I+sinh(x))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.26612, size = 88, normalized size = 1.8 \begin{align*} -\frac{6 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 7 i}{24 \,{\left (e^{x} - i\right )}^{3}} - \frac{-42 i \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} + 1015 i \, e^{\left (4 \, x\right )} - 1750 \, e^{\left (3 \, x\right )} - 1344 i \, e^{\left (2 \, x\right )} + 511 \, e^{x} + 79 i}{168 \,{\left (e^{x} + i\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="giac")

[Out]

-1/24*(6*I*e^(2*x) + 15*e^x - 7*I)/(e^x - I)^3 - 1/168*(-42*I*e^(6*x) + 315*e^(5*x) + 1015*I*e^(4*x) - 1750*e^
(3*x) - 1344*I*e^(2*x) + 511*e^x + 79*I)/(e^x + I)^7