### 3.218 $$\int \text{csch}(4 x) \sinh (x) \, dx$$

Optimal. Leaf size=26 $\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x))$

[Out]

-ArcTan[Sinh[x]]/4 + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])

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Rubi [A]  time = 0.0268189, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {1093, 203} $\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[4*x]*Sinh[x],x]

[Out]

-ArcTan[Sinh[x]]/4 + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \text{csch}(4 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{4+12 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{4+8 x^2} \, dx,x,\sinh (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{8+8 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{4} \tan ^{-1}(\sinh (x))+\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0214198, size = 26, normalized size = 1. $\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[4*x]*Sinh[x],x]

[Out]

-ArcTan[Sinh[x]]/4 + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])

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Maple [C]  time = 0.04, size = 62, normalized size = 2.4 \begin{align*}{\frac{i}{4}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{4}}\ln \left ({{\rm e}^{x}}+i \right ) +{\frac{i}{8}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{8}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(4*x)*sinh(x),x)

[Out]

1/4*I*ln(exp(x)-I)-1/4*I*ln(exp(x)+I)+1/8*I*2^(1/2)*ln(exp(2*x)+I*2^(1/2)*exp(x)-1)-1/8*I*2^(1/2)*ln(exp(2*x)-
I*2^(1/2)*exp(x)-1)

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Maxima [B]  time = 1.52082, size = 68, normalized size = 2.62 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) + \frac{1}{2} \, \arctan \left (e^{\left (-x\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x, algorithm="maxima")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-x))) - 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-x)))
+ 1/2*arctan(e^(-x))

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Fricas [B]  time = 2.18787, size = 297, normalized size = 11.42 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \cosh \left (x\right ) + \frac{1}{2} \, \sqrt{2} \sinh \left (x\right )\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \sqrt{2}}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - \frac{1}{2} \, \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(4*x)*sinh(x),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*arctan(1/2*sqrt(2)*cosh(x) + 1/2*sqrt(2)*sinh(x)) - 1/4*sqrt(2)*arctan(-1/2*(sqrt(2)*cosh(x)^2 + 2
*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))/(cosh(x) - sinh(x))) - 1/2*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 (4 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x)

[Out]

Integral(sinh(x)*csch(4*x), x)

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Giac [B]  time = 1.2097, size = 59, normalized size = 2.27 \begin{align*} -\frac{1}{8} \, \pi + \frac{1}{8} \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac{1}{4} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x, algorithm="giac")

[Out]

-1/8*pi + 1/8*sqrt(2)*(pi + 2*arctan(1/2*sqrt(2)*(e^(2*x) - 1)*e^(-x))) - 1/4*arctan(1/2*(e^(2*x) - 1)*e^(-x))