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

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

[Out]

1/2*arctan(sinh(x))

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Rubi [A]  time = 0.01, antiderivative size = 7, normalized size of antiderivative = 1.00, 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 3770

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \[ \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tanh[x/2]]

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fricas [A]  time = 0.42, size = 6, normalized size = 0.86 \[ \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]

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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giac [A]  time = 0.11, size = 3, normalized size = 0.43 \[ \arctan \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^x)

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maple [A]  time = 0.16, size = 4, normalized size = 0.57 \[ \arctan \left ({\mathrm e}^{x}\right ) \]

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 = 0.43, size = 7, normalized size = 1.00 \[ -\arctan \left (e^{\left (-x\right )}\right ) \]

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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mupad [B]  time = 0.05, size = 3, normalized size = 0.43 \[ \mathrm {atan}\left ({\mathrm {e}}^x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(exp(x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {csch}{\left (2 x \right )}\, dx \]

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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