### 3.628 $$\int (\text{sech}(x)+i \tanh (x)) \, dx$$

Optimal. Leaf size=11 $\tan ^{-1}(\sinh (x))+i \log (\cosh (x))$

[Out]

ArcTan[Sinh[x]] + I*Log[Cosh[x]]

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Rubi [A]  time = 0.008228, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {3770, 3475} $\tan ^{-1}(\sinh (x))+i \log (\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x] + I*Tanh[x],x]

[Out]

ArcTan[Sinh[x]] + I*Log[Cosh[x]]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (\text{sech}(x)+i \tanh (x)) \, dx &=i \int \tanh (x) \, dx+\int \text{sech}(x) \, dx\\ &=\tan ^{-1}(\sinh (x))+i \log (\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0044426, size = 17, normalized size = 1.55 $2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+i \log (\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[x] + I*Tanh[x],x]

[Out]

2*ArcTan[Tanh[x/2]] + I*Log[Cosh[x]]

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Maple [A]  time = 0., size = 11, normalized size = 1. \begin{align*} \arctan \left ( \sinh \left ( x \right ) \right ) +i\ln \left ( \cosh \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)+I*tanh(x),x)

[Out]

arctan(sinh(x))+I*ln(cosh(x))

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Maxima [A]  time = 1.01438, size = 12, normalized size = 1.09 \begin{align*} \arctan \left (\sinh \left (x\right )\right ) + i \, \log \left (\cosh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)+I*tanh(x),x, algorithm="maxima")

[Out]

arctan(sinh(x)) + I*log(cosh(x))

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Fricas [A]  time = 2.40346, size = 34, normalized size = 3.09 \begin{align*} -i \, x + 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),x, algorithm="fricas")

[Out]

-I*x + 2*I*log(e^x + I)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)+I*tanh(x),x)

[Out]

Integral(I*tanh(x) + sech(x), x)

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Giac [A]  time = 1.15644, size = 24, normalized size = 2.18 \begin{align*} -i \, x + 2 \, \arctan \left (e^{x}\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),x, algorithm="giac")

[Out]

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