3.636 \(\int (\text{sech}(x)-i \tanh (x))^3 \, dx\)

Optimal. Leaf size=28 \[ \frac{2 i}{1+i \sinh (x)}+i \log (-\sinh (x)+i) \]

[Out]

I*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])

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Rubi [A]  time = 0.0599885, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4391, 2667, 43} \[ \frac{2 i}{1+i \sinh (x)}+i \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Int[(Sech[x] - I*Tanh[x])^3,x]

[Out]

I*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])

Rule 4391

Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x_)]^(n_.))^(p_), x_Symbol] :> Int[A
ctivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0284044, size = 39, normalized size = 1.39 \[ -\frac{1}{2} i \tanh ^2(x)+\frac{3}{2} i \text{sech}^2(x)-\tan ^{-1}(\sinh (x))+i \log (\cosh (x))+2 \tanh (x) \text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sech[x] - I*Tanh[x])^3,x]

[Out]

-ArcTan[Sinh[x]] + I*Log[Cosh[x]] + ((3*I)/2)*Sech[x]^2 + 2*Sech[x]*Tanh[x] - (I/2)*Tanh[x]^2

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Maple [A]  time = 0.029, size = 45, normalized size = 1.6 \begin{align*} -{\rm sech} \left (x\right )\tanh \left ( x \right ) -2\,\arctan \left ({{\rm e}^{x}} \right ) -{\frac{{\frac{3\,i}{2}} \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}+3\,{\frac{\sinh \left ( x \right ) }{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}+i\ln \left ( \cosh \left ( x \right ) \right ) -{\frac{i}{2}} \left ( \tanh \left ( x \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sech(x)-I*tanh(x))^3,x)

[Out]

-sech(x)*tanh(x)-2*arctan(exp(x))-3/2*I*sinh(x)^2/cosh(x)^2+3/cosh(x)^2*sinh(x)+I*ln(cosh(x))-1/2*I*tanh(x)^2

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Maxima [B]  time = 1.57972, size = 99, normalized size = 3.54 \begin{align*} -\frac{3}{2} i \, \tanh \left (x\right )^{2} + i \, x + \frac{4 \,{\left (e^{\left (-x\right )} - e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \arctan \left (e^{\left (-x\right )}\right ) + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sech(x)-I*tanh(x))^3,x, algorithm="maxima")

[Out]

-3/2*I*tanh(x)^2 + I*x + 4*(e^(-x) - e^(-3*x))/(2*e^(-2*x) + e^(-4*x) + 1) + 2*I*e^(-2*x)/(2*e^(-2*x) + e^(-4*
x) + 1) + 2*arctan(e^(-x)) + I*log(e^(-2*x) + 1)

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Fricas [B]  time = 2.09119, size = 142, normalized size = 5.07 \begin{align*} \frac{-i \, x e^{\left (2 \, x\right )} - 2 \,{\left (x - 2\right )} e^{x} +{\left (2 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} - 2 i\right )} \log \left (e^{x} - i\right ) + i \, x}{e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sech(x)-I*tanh(x))^3,x, algorithm="fricas")

[Out]

(-I*x*e^(2*x) - 2*(x - 2)*e^x + (2*I*e^(2*x) + 4*e^x - 2*I)*log(e^x - I) + I*x)/(e^(2*x) - 2*I*e^x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - i \tanh ^{3}{\left (x \right )}\, dx - \int 3 \tanh ^{2}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx - \int 3 i \tanh{\left (x \right )} \operatorname{sech}^{2}{\left (x \right )}\, dx - \int - \operatorname{sech}^{3}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sech(x)-I*tanh(x))**3,x)

[Out]

-Integral(-I*tanh(x)**3, x) - Integral(3*tanh(x)**2*sech(x), x) - Integral(3*I*tanh(x)*sech(x)**2, x) - Integr
al(-sech(x)**3, x)

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Giac [A]  time = 1.15122, size = 28, normalized size = 1. \begin{align*} -i \, x + \frac{4 \, e^{x}}{{\left (e^{x} - i\right )}^{2}} + 2 i \, \log \left (e^{x} - i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sech(x)-I*tanh(x))^3,x, algorithm="giac")

[Out]

-I*x + 4*e^x/(e^x - I)^2 + 2*I*log(e^x - I)