### 3.216 $$\int \text{csch}(2 x) \sinh (x) \, dx$$

Optimal. Leaf size=7 $\frac{1}{2} \tan ^{-1}(\sinh (x))$

[Out]

ArcTan[Sinh[x]]/2

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Rubi [A]  time = 0.013489, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {4288, 3770} $\frac{1}{2} \tan ^{-1}(\sinh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[2*x]*Sinh[x],x]

[Out]

ArcTan[Sinh[x]]/2

Rule 4288

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

Rule 3770

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

Rubi steps

\begin{align*} \int \text{csch}(2 x) \sinh (x) \, dx &=\frac{1}{2} \int \text{sech}(x) \, dx\\ &=\frac{1}{2} \tan ^{-1}(\sinh (x))\\ \end{align*}

Mathematica [A]  time = 0.0028293, size = 7, normalized size = 1. $\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[2*x]*Sinh[x],x]

[Out]

ArcTan[Tanh[x/2]]

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Maple [A]  time = 0.013, size = 4, normalized size = 0.6 \begin{align*} \arctan \left ({{\rm e}^{x}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(2*x)*sinh(x),x)

[Out]

arctan(exp(x))

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Maxima [A]  time = 1.57405, size = 9, normalized size = 1.29 \begin{align*} -\arctan \left (e^{\left (-x\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*sinh(x),x, algorithm="maxima")

[Out]

-arctan(e^(-x))

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Fricas [A]  time = 2.07055, size = 36, normalized size = 5.14 \begin{align*} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*sinh(x),x, algorithm="fricas")

[Out]

arctan(cosh(x) + sinh(x))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*sinh(x),x)

[Out]

Integral(sinh(x)*csch(2*x), x)

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Giac [A]  time = 1.20529, size = 4, normalized size = 0.57 \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*sinh(x),x, algorithm="giac")

[Out]

arctan(e^x)